# Equations and Inequalities

## Linear Equations in One Variable

### Learning Objectives

1. Simplify expressions using order of operations (IA 1.1.3)
2. Solve linear equations using a general strategy (IA 2.1.1)

### Objective 1: Simplify expressions using order of operations (IA 1.1.3)

### Practice Makes Perfect

Evaluate the following expressions being sure to follow the order of operations:
Simplify by combining like terms:

### Objective 2: Solve linear equations using a general strategy (IA 2.1.1)

### Practice Makes Perfect

Solve each linear equation using the general strategy.
Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in .

### Solving Linear Equations in One Variable

A linear equation is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form
  and are solved using basic algebraic operations.
We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An identity equation is true for all values of the variable. Here is an example of an identity equation.
The solution set consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for
  will make the equation true.
A conditional equation is true for only some values of the variable. For example, if we are to solve the equation
  we have the following:
The solution set consists of one number:
  It is the only solution and, therefore, we have solved a conditional equation.
An inconsistent equation results in a false statement. For example, if we are to solve
  we have the following:
Indeed,
  There is no solution because this is an inconsistent equation.
Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.

### Solving a Rational Equation

In this section, we look at rational equations that, after some manipulation, result in a linear equation. If an equation contains at least one rational expression, it is a considered a rational equation.
Recall that a rational number is the ratio of two numbers, such as
  or
  A rational expression is the ratio, or quotient, of two polynomials. Here are three examples.
Rational equations have a variable in the denominator in at least one of the terms.
Our goal is to perform algebraic operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both sides of the equation by the least common denominator (LCD).
Finding the LCD is identifying an expression that contains the highest power of all of the factors in all of the denominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in each denominator will equal one and will cancel out.
A common mistake made when solving rational equations involves finding the LCD when one of the denominators is a binomial—two terms added or subtracted—such as
  Always consider a binomial as an individual factor—the terms cannot be separated. For example, suppose a problem has three terms and the denominators are

  and
  First, factor all denominators. We then have

  and
  as the denominators. (Note the parentheses placed around the second denominator.) Only the last two denominators have a common factor of
  The
  in the first denominator is separate from the
  in the
  denominators. An effective way to remember this is to write factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate factor. The LCD in this instance is found by multiplying together the
  one factor of
  and the 3. Thus, the LCD is the following:
So, both sides of the equation would be multiplied by
  Leave the LCD in factored form, as this makes it easier to see how each denominator in the problem cancels out.
Another example is a problem with two denominators, such as
  and
  Once the second denominator is factored as
  there is a common factor of x in both denominators and the LCD is

Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation.
We can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by crossing over the equal sign.
Multiply
  and
  which results in

Any solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.

### Finding a Linear Equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as
  where
  and
  Let us begin with the slope.

### The Slope of a Line

The slope of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.
If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in . The lines indicate the following slopes:

  and

### The Point-Slope Formula

Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.
This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

### Standard Form of a Line

Another way that we can represent the equation of a line is in standard form. Standard form is given as
where

  and
  are integers. The x- and y-terms are on one side of the equal sign and the constant term is on the other side.

### Vertical and Horizontal Lines

The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a vertical line is given as
where c is a constant. The slope of a vertical line is undefined, and regardless of the y-value of any point on the line, the x-coordinate of the point will be c.
Suppose that we want to find the equation of a line containing the following points:
  and
  First, we will find the slope.
Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through
  See .
The equation of a horizontal line is given as
where c is a constant. The slope of a horizontal line is zero, and for any x-value of a point on the line, the y-coordinate will be c.
Suppose we want to find the equation of a line that contains the following set of points:
  and
  We can use the point-slope formula. First, we find the slope using any two points on the line.
Use any point for
  in the formula, or use the y-intercept.
The graph is a horizontal line through
  Notice that all of the y-coordinates are the same. See .

### Determining Whether Graphs of Lines are Parallel or Perpendicular

Parallel lines have the same slope and different y-intercepts. Lines that are parallel to each other will never intersect. For example,  shows the graphs of various lines with the same slope,

All of the lines shown in the graph are parallel because they have the same slope and different y-intercepts.
Lines that are perpendicular intersect to form a
  -angle. The slope of one line is the negative reciprocal of the other. We can show that two lines are perpendicular if the product of the two slopes is
  For example,  shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of

### Writing the Equations of Lines Parallel or Perpendicular to a Given Line

As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use the point-slope formula to write the equation of the new line.

### Key Concepts

1. We can solve linear equations in one variable in the form
  using standard algebraic properties. See  and .
2. A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. See  and .
3. All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator. See  and  and .
4. Given two points, we can find the slope of a line using the slope formula. See .
5. We can identify the slope and y-intercept of an equation in slope-intercept form. See .
6. We can find the equation of a line given the slope and a point. See .
7. We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See .
8. The standard form of a line has no fractions. See .
9. Horizontal lines have a slope of zero and are defined as
  where c is a constant.
10. Vertical lines have an undefined slope (zero in the denominator), and are defined as
  where c is a constant. See .
11. Parallel lines have the same slope and different y-intercepts. See  and .
12. Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the equation for

For the following exercises, solve each rational equation for
  State all x-values that are excluded from the solution set.
For the following exercises, find the equation of the line using the point-slope formula.
Write all the final equations using the slope-intercept form.
For the following exercises, find the equation of the line using the given information.

### Graphical

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

### Numeric

For the following exercises, find the slope of the line that passes through the given points.
For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

### Technology

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

### Extensions

### Real-World Applications

For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be
  where
  is the number of miles traveled.


# Equations and Inequalities

## Models and Applications

### Learning Objectives

1. Solve a formula for a specified variable (IA 2.3.1)
2. Use a problem-solving strategy for word problems (IA 2.2.1)

### Objective 1: Solve a formula for a specified variable (IA 2.3.1)

### Practice Makes Perfect

Solve each formula for the specific variable.

### Objective 2: Use a problem-solving strategy for word problems (IA 2.2.1)

### Practice Makes Perfect

Use a Problem-Solving Strategy for word problems.
Neka is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only the final exam remains, and the maximum number of points that can be earned is 100. Is it possible for Neka to end the course with an A? A simple linear equation will give Neka his answer.
Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

### Setting up a Linear Equation to Solve a Real-World Application

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write
  This expression represents a variable cost because it changes according to the number of miles driven.
If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. We can use these quantities to model an equation that can be used to find the daily car rental cost

When dealing with real-world applications, there are certain expressions that we can translate directly into math.  lists some common verbal expressions and their equivalent mathematical expressions.

### Using a Formula to Solve a Real-World Application

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region,
  the perimeter of a rectangle,
  and the volume of a rectangular solid,
  When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

### Key Concepts

1. A linear equation can be used to solve for an unknown in a number problem. See .
2. Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. See .
3. There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the
  formula. See .
4. Many geometry problems are solved using the perimeter formula
  the area formula
  or the volume formula
  See , , and .

### Section Exercises

### Verbal

### Real-World Applications

For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked.
For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges of $.05/min for calls. Company B has a monthly fee of $5 and charges $.10/min for calls.
For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use
  for the number of devices that need data plans as part of their cost.
For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.
For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus $.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.
For the following exercises, use the formula given to solve for the required value.
For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.


# Equations and Inequalities

## Complex Numbers

### Learning Objectives

1. Use the product property to simplify radical expressions (IA 8.2.1)
2. Evaluate the square root of a negative number (IA 8.8.1)

### Objective 1: Use the product property to simplify radical expressions (IA 8.2.1)

### Practice Makes Perfect

Simplify a radical expression using the Product Property.

### Objective 2: Evaluate the square root of a negative number (IA 8.8.1)

Imaginary numbers result when we try to take the square root of a negative number. They do not belong to the set of real numbers and so are called imaginary or complex. We will see complex solutions when we solve quadratic equations whose graph does not touch the x-axis. Imaginary numbers are used in many real-life applications including the study of electricity involving alternating current (AC) electronics. Wireless, cellular and radar technologies utilize imaginary numbers. Electrical engineers use imaginary numbers to measure the amplitude and phase of electrical oscillations.

### Practice Makes Perfect

Evaluate the square root of a negative number. Perform any indicated operations and simplify.
Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.
In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.

### Expressing Square Roots of Negative Numbers as Multiples of

We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number
  is defined as the square root of

So, using properties of radicals,
We can write the square root of any negative number as a multiple of
  Consider the square root of

We use
  and not
  because the principal root of
  is the positive root.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written
  where
  is the real part and
  is the imaginary part. For example,
  is a complex number. So, too, is

Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.

### Plotting a Complex Number on the Complex Plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs
  where
  represents the coordinate for the horizontal axis and
  represents the coordinate for the vertical axis.
Let’s consider the number
  The real part of the complex number is
  and the imaginary part is 3. We plot the ordered pair
  to represent the complex number
  as shown in .

### Adding and Subtracting Complex Numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts.

### Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

### Multiplying a Complex Number by a Real Number

Lets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example,
  :

### Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term,
  it equals

### Dividing Complex Numbers

Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form
  We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of
  is
  For example, the product of
  and
  is
The result is a real number.
Note that complex conjugates have an opposite relationship: The complex conjugate of
  is
  and the complex conjugate of
  is
  Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to divide
  by
  where neither
  nor
  equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
Multiply the numerator and denominator by the complex conjugate of the denominator.
Apply the distributive property.
Simplify, remembering that

### Simplifying Powers of i

The powers of
  are cyclic. Let’s look at what happens when we raise
  to increasing powers.
We can see that when we get to the fifth power of
  it is equal to the first power. As we continue to multiply
  by increasing powers, we will see a cycle of four. Let’s examine the next four powers of

The cycle is repeated continuously:
  every four powers.

### Key Concepts

1. The square root of any negative number can be written as a multiple of
  See .
2. To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See .
3. Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See .
4. Complex numbers can be multiplied and divided.

5. The powers of
  are cyclic, repeating every fourth one. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, evaluate the algebraic expressions.

### Graphical

For the following exercises, plot the complex numbers on the complex plane.

### Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

### Technology

For the following exercises, use a calculator to help answer the questions.

### Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.


# Equations and Inequalities

## Quadratic Equations

### Learning Objectives

1. Recognize and use the appropriate method to factor a polynomial completely (IA 6.4.1)

### Objective 1: Recognize and use the appropriate method to factor a polynomial completely  (IA 6.4.1)

The following chart summarizes the factoring methods and outlines a strategy you should use when factoring polynomials.

### Practice Makes Perfect

Recognize and use the appropriate method to factor a polynomial completely. Factor each of the following polynomials completely, if a polynomial does not factor label it as prime.

The computer monitor on the left in  is a 23.6-inch model and the one on the right is a 27-inch model. Proportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods.

### Solving Quadratic Equations by Factoring

An equation containing a second-degree polynomial is called a quadratic equation. For example, equations such as
  and
  are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.
Often the easiest method of solving a quadratic equation is factoring. Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation.
If a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the zero-product property, which states that if
  then
  or
  where a and b are real numbers or algebraic expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero.
Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression
  by multiplying the two factors together.
The product is a quadratic expression. Set equal to zero,
  is a quadratic equation. If we were to factor the equation, we would get back the factors we multiplied.
The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form,
  where a, b, and c are real numbers, and
  The equation
  is in standard form.
We can use the zero-product property to solve quadratic equations in which we first have to factor out the greatest common factor (GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section.

### Solving Quadratics with a Leading Coefficient of 1

In the quadratic equation
  the leading coefficient, or the coefficient of
  is 1. We have one method of factoring quadratic equations in this form.

### Solving a Quadratic Equation by Factoring when the Leading Coefficient is not 1

When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let’s review the grouping procedures:
1. With the quadratic in standard form,
  multiply

2. Find two numbers whose product equals
  and whose sum equals

3. Rewrite the equation replacing the
  term with two terms using the numbers found in step 1 as coefficients of x.
4. Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
5. Factor out the expression in parentheses.
6. Set the expressions equal to zero and solve for the variable.

### Using the Square Root Property

When there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property, in which we isolate the
  term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the
  term so that the square root property can be used.

### Completing the Square

Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation known as completing the square. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a, must equal 1. If it does not, then divide the entire equation by a. Then, we can use the following procedures to solve a quadratic equation by completing the square.
We will use the example
  to illustrate each step.
1. Given a quadratic equation that cannot be factored, and with

2. Multiply the

3.
Add

4. The left side of the equation can now be factored as a perfect square.

5. Use the square root property and solve.

6. The solutions are

### Using the Quadratic Formula

The fourth method of solving a quadratic equation is by using the quadratic formula, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.
We can derive the quadratic formula by completing the square. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by
  and obtain a positive a. Given

  we will complete the square as follows:
1. First, move the constant term to the right side of the equal sign:

2. As we want the leading coefficient to equal 1, divide through by

3. Then, find

4. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:

5. Now, use the square root property, which gives

6. Finally, add

### The Discriminant

The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant, or the expression under the radical,
  The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect.  relates the value of the discriminant to the solutions of a quadratic equation.

### Using the Pythagorean Theorem

One of the most famous formulas in mathematics is the Pythagorean Theorem. It is based on a right triangle, and states the relationship among the lengths of the sides as
  where
  and
  refer to the legs of a right triangle adjacent to the
  angle, and
  refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.
We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.
The Pythagorean Theorem is given as
where
  and
  refer to the legs of a right triangle adjacent to the
  angle, and
  refers to the hypotenuse, as shown in .

### Key Equations

### Key Concepts

1. Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-product property is then used to find solutions. See , , and .
2. Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See  and .
3. Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See  and .
4. Completing the square is a method of solving quadratic equations when the equation cannot be factored. See .
5. A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See  and .
6. The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See .
7. The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the quadratic equation by factoring.
For the following exercises, solve the quadratic equation by using the square root property.
For the following exercises, solve the quadratic equation by completing the square. Show each step.
For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.
For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

### Technology

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

### Extensions

### Real-World Applications


# Equations and Inequalities

## Other Types of Equations

We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.

### Solving Equations Involving Rational Exponents

Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example,
  is another way of writing

  is another way of writing
  The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.
We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example,

  and so on.

### Solving Equations Using Factoring

We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.

### Solving Radical Equations

Radical equations are equations that contain variables in the radicand (the expression under a radical symbol), such as
Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find extraneous solutions, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.

### Solving an Absolute Value Equation

Next, we will learn how to solve an absolute value equation. To solve an equation such as
  we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is
  or
  This leads to two different equations we can solve independently.
Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

### Solving Other Types of Equations

There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.

### Solving Equations in Quadratic Form

Equations in quadratic form are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include
  and
  In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.

### Solving Rational Equations Resulting in a Quadratic

Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.

### Key Concepts

1. Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See , , and .
2. Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See  and .
3. We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index. See  and .
4. To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value. See .
5. Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve. See  and .
6. Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the rational exponent equation. Use factoring where necessary.
For the following exercises, solve the following polynomial equations by grouping and factoring.
For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.
For the following exercises, solve the equation involving absolute value.
For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

### Extensions

For the following exercises, solve for the unknown variable.

### Real-World Applications

For the following exercises, use the model for the period of a pendulum,
  such that
  where the length of the pendulum is L and the acceleration due to gravity is

For the following exercises, use a model for body surface area, BSA, such that
  where w = weight in kg and h = height in cm.


# Equations and Inequalities

## Linear Inequalities and Absolute Value Inequalities

It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.

### Using Interval Notation

Indicating the solution to an inequality such as
    can be achieved in several ways.
We can use a number line as shown in . The blue ray begins at
    and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.
We can use set-builder notation:
    which translates to “all real numbers x such that x is greater than or equal to 4.” Notice that braces are used to indicate a set.
The third method is interval notation, in which solution sets are indicated with parentheses or brackets. The solutions to
    are represented as
    This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.
The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are
    or all numbers between
    and
    including
    but not including

    all real numbers between, but not including
    and
    and
    all real numbers less than and including
     outlines the possibilities.

### Using the Properties of Inequalities

When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.

### Solving Inequalities in One Variable Algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.

### Understanding Compound Inequalities

A compound inequality includes two inequalities in one statement. A statement such as
    means
    and
    There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.

### Solving Absolute Value Inequalities

As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at
    has an absolute value of
    as it is x units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.
An absolute value inequality is an equation of the form
Where A, and sometimes B, represents an algebraic expression dependent on a variable x. Solving the inequality means finding the set of all
    -values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of x-values such that the distance between
    and 600 is less than or equal to 200. We represent the distance between
    and 600 as
    and therefore,
    or
This means our returns would be between $400 and $800.
To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.

### Key Concepts

1. Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. See  and .
2. Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. See , , , and .
3. Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. See  and .
4. Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See  and .
5. Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the inequality. Write your final answer in interval notation.
For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.
For the following exercises, describe all the x-values within or including a distance of the given values.
For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.

### Graphical

For the following exercises, graph the function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.
For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines.

### Numeric

For the following exercises, write the set in interval notation.
For the following exercises, write the interval in set-builder notation.
For the following exercises, write the set of numbers represented on the number line in interval notation.

### Technology

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2nd CALC 5:intersection, 1st curve, enter, 2nd curve, enter, guess, enter). Copy a sketch of the graph and shade the x-axis for your solution set to the inequality. Write final answers in interval notation.

### Extensions

### Real-World Applications

### Chapter Review Exercises

### The Rectangular Coordinate Systems and Graphs

For the following exercises, find the x-intercept and the y-intercept without graphing.
For the following exercises, solve for y in terms of x, putting the equation in slope–intercept form.
For the following exercises, find the distance between the two points.
For the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.
For the following exercises, construct a table and graph the equation by plotting at least three points.

### Linear Equations in One Variable

For the following exercises, solve for

For the following exercises, solve for
    State all x-values that are excluded from the solution set.
For the following exercises, find the equation of the line using the point-slope formula.

### Models and Applications

For the following exercises, write and solve an equation to answer each question.

### Complex Numbers

For the following exercises, use the quadratic equation to solve.
For the following exercises, name the horizontal component and the vertical component.
For the following exercises, perform the operations indicated.

### Quadratic Equations

For the following exercises, solve the quadratic equation by factoring.
For the following exercises, solve the quadratic equation by using the square-root property.
For the following exercises, solve the quadratic equation by completing the square.
For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution.
For the following exercises, solve the quadratic equation by the method of your choice.

### Other Types of Equations

For the following exercises, solve the equations.

### Linear Inequalities and Absolute Value Inequalities

For the following exercises, solve the inequality. Write your final answer in interval notation.
For the following exercises, solve the compound inequality. Write your answer in interval notation.
For the following exercises, graph as described.

### Chapter Practice Test

For the following exercises, find the equation of the line with the given information.
For the following exercises, find the real solutions of each equation by factoring.


# Functions

## Introduction to Functions

Toward the end of the twentieth century, the values of stocks of Internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. The graph above tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.
Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.


# Functions

## Functions and Function Notation

### Learning Objectives

1. Find the value of a function (IA 3.5.3)

### Objective 1: Find the value of a function (IA 3.5.3)

A relation is any set of ordered pairs, (x,y). The collection of x-values in the ordered pairs together make up the domain. The collection of y-values in the ordered pairs together make up the range.
A special type of relation, called a function, is studied extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.

### Practice Makes Perfect

Find the value of a function
A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

### Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.
The domain is

 The range is

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter
  Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter

A function
  is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain,
  is paired with exactly one element in the range,

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

Notice that each element in the domain,

 is not paired with exactly one element in the range,

For example, the term “odd” corresponds to three values from the range,

 and the term “even” corresponds to two values from the range,

 This violates the definition of a function, so this relation is not a function.
 compares relations that are functions and not functions.

### Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.
To represent “height is a function of age,” we start by identifying the descriptive variables

 for height and

 for age. The letters
  and
  are often used to represent functions just as we use
  and
  to represent numbers and

 and
  to represent sets.

Remember, we can use any letter to name the function; the notation
 shows us that
 depends on
  The value
 must be put into the function
 to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.
We can also give an algebraic expression as the input to a function. For example
 means “first add a and b, and the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.

### Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.
 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function
 where
 identifies months by an integer rather than by name.
 defines a function
 Remember, this notation tells us that
 is the name of the function that takes the input
 and gives the output

 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

### Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.
When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

### Evaluation of Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function
 can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

### Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation
 expresses a functional relationship between

and
  We can rewrite it to decide if
  is a function of

### Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.
The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See .http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.
At times, evaluating a function in table form may be more useful than using equations.Here let us call the function

 The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function
 at the input value of “goldfish.” We would write
  Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function
 seems ideally suited to this function, more so than writing it in paragraph or function form.

### Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

### Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.
However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in .
This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.
To visualize this concept, let’s look again at the two simple functions sketched in (a) and (b). The function in part (a) shows a relationship that is not a one-to-one function because inputs
  and
  both give output
  The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

### Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.
The most common graphs name the input value
  and the output value
  and we say
  is a function of
  or
  when the function is named
  The graph of the function is the set of all points
  in the plane that satisfies the equation
  If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in  tell us that
  and
  However, the set of all points
  satisfying
  is a curve. The curve shown includes
  and
  because the curve passes through those points.
The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See .

###  Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

### Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use
  as the input variable and
  as the output variable.
We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in .

### Key Equations

### Key Concepts

1. A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See  and .
2. Function notation is a shorthand method for relating the input to the output in the form
 See  and .
3. In tabular form, a function can be represented by rows or columns that relate to input and output values. See .
4. To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See  and .
5. To solve for a specific function value, we determine the input values that yield the specific output value. See .
6. An algebraic form of a function can be written from an equation. See  and .
7. Input and output values of a function can be identified from a table. See .
8. Relating input values to output values on a graph is another way to evaluate a function. See .
9. A function is one-to-one if each output value corresponds to only one input value. See .
10. A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See .
11. The graph of a one-to-one function passes the horizontal line test. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the relation represents a function.
For the following exercises, determine whether the relation represents
 as a function of

For the following exercises, evaluate

### Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.
For the following exercises, determine if the given graph is a one-to-one function.

### Numeric

For the following exercises, determine whether the relation represents a function.
For the following exercises, determine if the relation represented in table form represents
 as a function of

For the following exercises, use the function
 represented in the table below.
For the following exercises, evaluate the function
 at the values
 and

For the following exercises, evaluate the expressions, given functions
 and

### Technology

For the following exercises, graph
  on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
 on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
 on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
 on the given domain. Determine the corresponding range. Show each graph.

###  Real-World Applications


# Functions

## Domain and Range

### Learning Objectives

1. Find the domain and range of a function (IA 3.5.1)
A relation is any set of ordered pairs, (x,y). A special type of relation, called a function, is studied extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.
When studying functions, it’s important to be able to identify potential input values, called the domain, and potential output values, called the range.
A graph of a function can always help in identifying domain and range. When graphing basic functions, we can scan the x-axis just as we read in English from left to right to help determine the domain. We will scan the y-axis from bottom to top to help determine the range. So, in finding both domain and range, we scan axes from smallest to largest to see which values are defined. Typically, we will use interval notation, where you show the endpoints of defined sets using parentheses (endpoint not included) or brackets (endpoint is included) to express both the domain, D, and range, R, of a relation or function.
When working with functions expressed as an equation, the following steps can help to identify the domain.

### Activity: Restrictions on the domain of functions

Without a calculator, complete the following:

  _____;  _____;  _____;  _____;  _____;  _____;  _____
Clearly describe the two “trouble spots” which prevent expressions from representing real numbers:
1. ________________________________________________________________________________
2. ________________________________________________________________________________
3. Keeping these trouble spots in mind, algebraically determine the domain of each function.  Write each answer in interval notation below the function.  Remember, looking at the graph of the function can always help in finding domain and range.

### Practice Makes Perfect

Find the domain and range of a function.
Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring.  shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

### Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.
We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See .
We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write
 We will discuss interval notation in greater detail later.
Let’s turn our attention to finding the domain of a function whose equation is provided.  Oftentimes, finding the domain of such functions involves remembering three different forms.  First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers.  Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero.  Third, if there is an even root, consider excluding values that would make the radicand negative.
Before we begin, let us review the conventions of interval notation:
1. The smallest number from the interval is written first.
2. The largest number in the interval is written second, following a comma.
3. Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.
4. Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.
See  for a summary of interval notation.

### Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation. For example,
 describes the behavior of
 in set-builder notation. The braces
 are read as “the set of,” and the vertical bar | is read as “such that,” so we would read
 as “the set of x-values such that 10 is less than or equal to
 and
 is less than 30.”
 compares inequality notation,  set-builder notation, and interval notation.
To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol,
 to combine two unconnected intervals. For example, the union of the sets

and

is the set
 It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

### Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See .
We can observe that the graph extends horizontally from
 to the right without bound, so the domain is
 The vertical extent of the graph is all range values
  and below, so the range is
 Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

### Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

### Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function
 With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.
If we input 0, or a positive value, the output is the same as the input.
If we input a negative value, the output is the opposite of the input.
Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.
We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income
 would be
 if
  and
  if

### Key Concepts

1. The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
2. The domain of a function can be determined by listing the input values of a set of ordered pairs. See .
3. The domain of a function can also be determined by identifying the input values of a function written as an equation. See , , and .
4. Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See .
5. For many functions, the domain and range can be determined from a graph. See  and .
6. An understanding of toolkit functions can be used to find the domain and range of related functions. See , , and .
7. A piecewise function is described by more than one formula. See  and .
8. A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the domain of each function using interval notation.

### Graphical

For the following exercises, write the domain and range of each function using interval notation.
For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

### Numeric

For the following exercises, given each function
 evaluate
 and

For the following exercises, given each function
 evaluate
 and

For the following exercises, write the domain for the piecewise function in interval notation.

### Technology

###  Extension

###  Real-World Applications


# Functions

## Rates of Change and Behavior of Graphs

### Learning Objectives

1. Find the slope of a line (IA 3.2.1)

### Objective 1: Find the slope of a line (IA 3.2.1)

In our work with functions we will make observations about when the function increases or decreases and how quickly this change takes place. The average rate of change is a measure of change of a function and tells us how an output quantity, or y value, changes relative to an input quantity, or x value. Finding the average rate of change between two points is equivalent to finding the slope of the line segment connecting these two data points.

When interpreting an average rate of change it will be important to consider the units of measurement. Make sure to always attach these units to both the numerator and denominator when they are provided to you.

### Activity: Finding slopes of lines.

          ⓐ Which pair of lines appear parallel?  ________ and ________.
          Find their slopes: ________;  _________

          ⓑ Which pair of lines appear perpendicular?  ________ and ________.
Find their slopes: ________;  _________

          ⓒ Complete the following:
Two lines are parallel if their slopes are _________.
Two lines are perpendicular if their slopes are _________.

### Practice Makes Perfect

Find the slope of a line and the average rate of change.

Gasoline costs have experienced some wild fluctuations over the last several decades. http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524. Accessed 3/5/2014. lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.
If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year. In this section, we will investigate changes such as these.

### Finding the Average Rate of Change of a Function

The price change per year is a rate of change because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in  did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.
The Greek letter
  (delta) signifies the change in a quantity; we read the ratio as “delta-y over delta-x” or “the change in
    divided by the change in
   ” Occasionally we write
  instead of
  which still represents the change in the function’s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.
In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was
On average, the price of gas increased by about 19.6¢ each year.
Other examples of rates of change include:
1. A population of rats increasing by 40 rats per week
2. A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)
3. A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)
4. The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage
5. The amount of money in a college account decreasing by $4,000 per quarter

### Using a Graph to Determine Where a Function is Increasing, Decreasing, or Constant

As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative.  shows examples of increasing and decreasing intervals on a function.
While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is the location of a  local maximum. The function value at that point is the local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is the location of a local minimum. The function value at that point is the local minimum. The plural form is “local minima.” Together, local maxima and minima are called local extrema, or local extreme values, of the function. (The singular form is “extremum.”) Often, the term local is replaced by the term relative. In this text, we will use the term local.
Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.
For the function whose graph is shown in , the local maximum is 16, and it occurs at
  The local minimum is
    and it occurs at

To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points.  illustrates these ideas for a local maximum.
These observations lead us to a formal definition of local extrema.

### Analyzing the Toolkit Functions for Increasing or Decreasing Intervals

We will now return to our toolkit functions and discuss their graphical behavior in , , and .

###  Use A Graph to Locate the Absolute Maximum and Absolute Minimum

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The
  coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively.
To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See .
Not every function has an absolute maximum or minimum value. The toolkit function
    is one such function.

### Key Equations

### Key Concepts

1. A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See .
2. Identifying points that mark the interval on a graph can be used to find the average rate of change. See .
3. Comparing pairs of input and output values in a table can also be used to find the average rate of change. See .
4. An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See  and .
5. The average rate of change can sometimes be determined as an expression. See .
6. A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See .
7. A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
8. A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
9. Minima and maxima are also called extrema.
10. We can find local extrema from a graph. See  and .
11. The highest and lowest points on a graph indicate the maxima and minima. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the average rate of change of each function on the interval specified for real numbers
     or
     in simplest form.

### Graphical

For the following exercises, consider the graph of
     shown in .
For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.
For the following exercises, consider the graph shown in .
For the following exercises, consider the graph in .

### Numeric

For the following exercises, find the average rate of change of each function on the interval specified.

### Technology

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

### Extension

### Real-World Applications


# Functions

## Composition of Functions

### Learning Objectives

1. Find the value of a function (IA 3.5.3), (CA 3.1.2)

### Objective 1: Find the value of a function (IA 3.5.3), (CA 3.1.2)

A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each  -value is matched with only one  -value.
The notation   defines a function named  .  This is read as “   is a function of  .” The letter   represents the input value, or independent variable. The letter , or  , represents the output value, or dependent variable.

### Practice Makes Perfect

Find the value of a function.

A composite function is a two-step function and can have numerical or variable inputs.

 is read as “f of g of x”
To evaluate a composite function, we always start evaluating the inner function and then evaluate the outer function in terms of the inner function.
Let’s use a table to help us organize our work in evaluating a two-step (composition) function in terms of some numerical inputs.
First evaluate g in terms of x, the f in terms of g(x).
Given that:  , and  , complete the table below. Remember the output of g(x) becomes the input of f(x)!

### Practice Makes Perfect

Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.
Using descriptive variables, we can notate these two functions. The function
 gives the cost
 of heating a house for a given average daily temperature in
 degrees Celsius. The function
 gives the average daily temperature on day
 of the year. For any given day,
 means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature
 For example, we could evaluate
 to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the cost function at that temperature. We would write

By combining these two relationships into one function, we have performed function composition, which is the focus of this section.

### Combining Functions Using Algebraic Operations

Function composition is only one way to combine existing functions. Another way is to carry out the usual algebraic operations on functions, such as addition, subtraction, multiplication and division. We do this by performing the operations with the function outputs, defining the result as the output of our new function.
Suppose we need to add two columns of numbers that represent a husband and wife’s separate annual incomes over a period of years, with the result being their total household income. We want to do this for every year, adding only that year’s incomes and then collecting all the data in a new column. If
 is the wife’s income and
 is the husband’s income in year
 and we want
 to represent the total income, then we can define a new function.
If this holds true for every year, then we can focus on the relation between the functions without reference to a year and write
Just as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that have the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units when we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.
For two functions
 and
 with real number outputs, we define new functions
 and
 by the relations

### Create a Function by Composition of Functions

Performing algebraic operations on functions combines them into a new function, but we can also create functions by composing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that takes a day as input and yields a cost as output. The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation:
We read the left-hand side as
  composed with
 at
  and the right-hand side as
  of
 of
  The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol
 is called the composition operator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves without referring to any particular input value. Composition is a binary operation that takes two functions and forms a new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important not to confuse function composition with multiplication because, as we learned above, in most cases

It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the equation above, the function
 takes the input
 first and yields an output
 Then the function
 takes
 as an input and yields an output

In general,
 and
 are different functions. In other words, in many cases
 for all
 We will also see that sometimes two functions can be composed only in one specific order.
For example, if
 and
 then
but
These expressions are not equal for all values of
 so the two functions are not equal. It is irrelevant that the expressions happen to be equal for the single input value

Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.

### Evaluating Composite Functions

Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.

### Evaluating Composite Functions Using Tables

When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.

### Evaluating Composite Functions Using Graphs

When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the
 and
 axes of the graphs.

### Evaluating Composite Functions Using Formulas

When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.
While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition
 To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like
 we substitute the value inside the parentheses into the formula wherever we see the input variable.

### Finding the Domain of a Composite Function

As we discussed previously, the domain of a composite function such as
 is dependent on the domain of
 and the domain of
 It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as
 Let us assume we know the domains of the functions
 and
 separately. If we write the composite function for an input
 as
 we can see right away that
 must be a member of the domain of
 in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that
 must be a member of the domain of
 otherwise the second function evaluation in
 cannot be completed, and the expression is still undefined. Thus the domain of
 consists of only those inputs in the domain of
 that produce outputs from
 belonging to the domain of
 Note that the domain of
 composed with
 is the set of all
 such that
 is in the domain of
 and
 is in the domain of

### Decomposing a Composite Function into its Component Functions

In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function, so we may choose the decomposition that appears to be most expedient.

### Key Equation

### Key Concepts

1. We can perform algebraic operations on functions. See .
2. When functions are composed, the output of the first (inner) function becomes the input of the second (outer) function.
3. The function produced by composing two functions is a composite function. See  and .
4. The order of function composition must be considered when interpreting the meaning of composite functions. See .
5. A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
6. A composite function can be evaluated from a table. See .
7. A composite function can be evaluated from a graph. See .
8. A composite function can be evaluated from a formula. See .
9. The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See  and .
10. Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
11. Functions can often be decomposed in more than one way. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine the domain for each function in interval notation.
For the following exercises, use each pair of functions to find
 and
 Simplify your answers.
For the following exercises, use each set of functions to find
 Simplify your answers.
For the following exercises, find functions
 and
 so the given function can be expressed as

### Graphical

For the following exercises, use the graphs of
 shown in , and
 shown in , to evaluate the expressions.
For the following exercises, use graphs of
 shown in ,
 shown in , and
 shown in , to evaluate the expressions.

### Numeric

For the following exercises, use the function values for
 shown in  to evaluate each expression.
For the following exercises, use the function values for
 shown in  to evaluate the expressions.
For the following exercises, use each pair of functions to find
 and

For the following exercises, use the functions
 and

to evaluate or find the composite function as indicated.

### Extensions

For the following exercises, use
 and

For the following exercises, let

 and

For the following exercises, find the composition when
 for all
 and

### Real-World Applications