# 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


# Functions

## Transformation of Functions

### Learning Objectives

1. Identify graphs of basic functions, (IA 3.6.2)
2. Graph quadratic functions using transformations, (IA 9.7.4)

### Objective 1: Identify graphs of basic functions, (IA 3.6.2)

Basic functions have unique shapes, characteristics, and algebraic equations. It will be helpful to recognize and identify these basic or “toolkit functions” in our work in algebra, precalculus and calculus. Remember functions can be represented in many ways including by name, equation, graph, and basic tables of values.

### Practice Makes Perfect

Use a graphing program to help complete the following.  Then, choose three values of x to evaluate for each.  Add the x and y to the table for each exercise.

### Objective 2:  Graph quadratic functions using transformations (IA 9.7.4)

When we modify basic functions by adding, subtracting, or multiplying constants to the equation, very systematic changes take place. We call these transformations of basic functions. Here we will investigate the effects of vertical shifts, horizontal shifts, vertical stretches or compressions, and reflections on quadratic functions. We could use any basic function to illustrate transformations, but quadratics work nicely because we can easily keep track of a point called the vertex.

### Practice Makes Perfect

The graphs of quadratic functions are called parabolas. Use a graphing program to graph each of the following quadratic functions. For each graph find the vertex (the minimum or maximum value) of the parabola and list its coordinates. Most importantly use the patterns observed to answer each of the given questions.
We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.

### Graphing Functions Using Vertical and Horizontal Shifts

Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.

### Identifying Vertical Shifts

One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function
   the function
   is shifted vertically
   units. See  for an example.
To help you visualize the concept of a vertical shift, consider that
  Therefore,
  is equivalent to
  Every unit of
  is replaced by
  so the y-value increases or decreases depending on the value of
  The result is a shift upward or downward.

### Identifying Horizontal Shifts

We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift, shown in .
For example, if
  then
  is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in

###  Combining Vertical and Horizontal Shifts

Now that we have two transformations, we can combine them. Vertical shifts are outside changes that affect the output (y-) values and shift the function up or down. Horizontal shifts are inside changes that affect the input (x-) values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down and left or right.

### Graphing Functions Using Reflections about the Axes

Another transformation that can be applied to a function is a reflection over the x- or y-axis. A vertical reflection reflects a graph vertically across the x-axis, while a horizontal reflection reflects a graph horizontally across the y-axis. The reflections are shown in .
Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y-axis.

### Determining Even and Odd Functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions
    or
    will result in the original graph. We say that these types of graphs are symmetric about the y-axis. A function whose graph is symmetric about the y-axis is called an even function.
If the graphs of
    or
    were reflected over both axes, the result would be the original graph, as shown in .
We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function.
Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example,
    is neither even nor odd. Also, the only function that is both even and odd is the constant function

### Graphing Functions Using Stretches and Compressions

Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.
We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.

###  Vertical Stretches and Compressions

When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression.  shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.

###  Horizontal Stretches and Compressions

Now we consider changes to the inside of a function. When we multiply a function’s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a horizontal stretch; if the constant is greater than 1, we get a horizontal compression of the function.
Given a function
   the form
   results in a horizontal stretch or compression. Consider the function
   Observe . The graph of
   is a horizontal stretch of the graph of the function
   by a factor of 2. The graph of
   is a horizontal compression of the graph of the function
   by a factor of .

### Performing a Sequence of Transformations

When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.
When we see an expression such as
  which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of
    we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.
Horizontal transformations are a little trickier to think about. When we write
    for example, we have to think about how the inputs to the function
    relate to the inputs to the function
    Suppose we know
    What input to
    would produce that output? In other words, what value of
    will allow
    We would need
    To solve for
    we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.
This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.
Let’s work through an example.
We can factor out a 2.
Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.

### Key Equations

### Key Concepts

1. A function can be shifted vertically by adding a constant to the output. See  and .
2. A function can be shifted horizontally by adding a constant to the input. See , , and .
3. Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See .
4. Vertical and horizontal shifts are often combined. See  and .
5. A vertical reflection reflects a graph about the
  axis. A graph can be reflected vertically by multiplying the output by –1.
6. A horizontal reflection reflects a graph about the
  axis. A graph can be reflected horizontally by multiplying the input by –1.
7. A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See .
8. A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See .
9. A function presented as an equation can be reflected by applying transformations one at a time. See .
10. Even functions are symmetric about the
  axis, whereas odd functions are symmetric about the origin.
11. Even functions satisfy the condition

12. Odd functions satisfy the condition

13. A function can be odd, even, or neither. See .
14. A function can be compressed or stretched vertically by multiplying the output by a constant. See , , and .
15. A function can be compressed or stretched horizontally by multiplying the input by a constant. See , , and .
16. The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, write a formula for the function obtained when the graph is shifted as described.
For the following exercises, describe how the graph of the function is a transformation of the graph of the original function

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

### Graphical

For the following exercises, use the graph of
   shown in  to sketch a graph of each transformation of

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

### Numeric

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.
For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.
For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.
For the following exercises, determine whether the function is odd, even, or neither.
For the following exercises, describe how the graph of each function is a transformation of the graph of the original function

For the following exercises, write a formula for the function
   that results when the graph of a given toolkit function is transformed as described.
For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.
For the following exercises, use the graph in  to sketch the given transformations.


# Functions

## Absolute Value Functions

### Learning Objectives

1. Solve absolute value equations (IA 2.7.1)
2. Identify graphs of absolute value functions (IA 3.6.2)

### Objective 1: Solve absolute value equations (IA 2.7.1)

Recall that in its basic form, 𝑓(𝑥)=|𝑥|, the absolute value function is one of our toolkit functions. The absolute value function is often thought of as providing the distance the number is from zero on a number line. Numerically, for whatever the input value is, the output is the magnitude of this value.

The absolute value function can be defined as a piecewise function
  , when  or  , when

### Practice Makes Perfect

Solve absolute value equations.

### Objective 2: Identify and graph absolute value functions (IA 3.6.2)

Absolute value functions have a “V” shaped graph. If scanning this function from left to right the corner is the point where the graph changes direction.

  , when  or  , when

### Practice Makes Perfect

Identify and graph absolute value functions. Graph each of the following functions. Label at least one point on your graph.
Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will continue our investigation of absolute value functions.

### Understanding Absolute Value

Recall that in its basic form
    the absolute value function is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value functions to solve some kinds of real-world problems.

### Graphing an Absolute Value Function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in .
 shows the graph of
  The graph of
  has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at
    for this transformed function.

### Solving an Absolute Value Equation

In Other Type of Equations, we touched on the concepts of absolute value equations. Now that we understand a little more about their graphs, we can take another look at these types of equations. Now that we can graph an absolute value function, 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 is 8 or -8. 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.
An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

### Key Concepts

1. Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See .
2. The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See .
3. In an absolute value equation, an unknown variable is the input of an absolute value function.
4. If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the x- and y-intercepts of the graphs of each function.

### Graphical

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.
For the following exercises, graph the given functions by hand.

### Technology

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

### Extensions

For the following exercises, solve the inequality.

### Real-World Applications


# Functions

## Inverse Functions

### Learning Objectives

1. Find and evaluate composite functions (IA 10.1.1).
2. Determine whether a function is one-to-one (IA 10.1.2).

### Objective 1: Find and evaluate composite functions (IA 10.1.1).

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 by evaluating the inner function and then evaluate the outer function in terms of the inner function.

### Practice Makes Perfect

Find and evaluate composite functions.

For each of the following function pairs find:

### Objective 2: Determine whether a function is one-to-one (IA 10.1.2).

In creating a process called a function, f(x), it is often useful to undo this process, or create an inverse to the function, f-1(x). When finding the inverse, we restrict our work to one-to-one functions, this means that the inverse we find should also be one-to-one. Remember that the horizontal line test is a great way to check to see if a graph represents a one-to-one function.

For any one-to-one function f(x), the inverse is a function f-1(x) such that  and  .
The following key terms will be important to our understanding of functions and their inverses.
Function: a relation in which each input value yields a unique output value.
Vertical line test: a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once.
One-to-one function: a function for which each value of the output is associated with a unique input value.
Horizontal line test: a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once.

### Practice Makes Perfect

Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.
If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards.  provides a visual representation of this question. In this section, we will consider the reverse nature of functions.

###  Verifying That Two Functions Are Inverse Functions

Betty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees Fahrenheit to degrees Celsius using the formula
and substitutes 75 for
   to calculate
Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, Betty gets the week’s weather forecast from  for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.
At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for
   after substituting a value for
   For example, to convert 26 degrees Celsius, she could write
After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.
The formula for which Betty is searching corresponds to the idea of an inverse function, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.
Given a function
   we represent its inverse as
   read as
   inverse of
   The raised
   is part of the notation. It is not an exponent; it does not imply a power of
   . In other words,
   does not mean
   because
   is the reciprocal of
   and not the inverse.
The “exponent-like” notation comes from an analogy between function composition and multiplication: just as
   (1 is the identity element for multiplication) for any nonzero number
   so
   equals the identity function, that is,
This holds for all
   in the domain of
   Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal, some functions do not have inverses.
Given a function
   we can verify whether some other function
   is the inverse of
   by checking whether either
   or
   is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)
For example,
   and
   are inverse functions.
and
A few coordinate pairs from the graph of the function
   are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function
   are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

### Finding Domain and Range of Inverse Functions

The outputs of the function
   are the inputs to
   so the range of
   is also the domain of
   Likewise, because the inputs to
   are the outputs of
   the domain of
   is the range of
   We can visualize the situation as in .
When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of
   is
   because a square “undoes” a square root; but the square is only the inverse of the square root on the domain
   since that is the range of

We can look at this problem from the other side, starting with the square (toolkit quadratic) function
   If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and –3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the “inverse” is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.
In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function
   with its domain limited to
   which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).
If
   on
   then the inverse function is

1. The domain of
   = range of
   =

2. The domain of
   = range of
   =

###  Finding and Evaluating Inverse Functions

Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.

### Inverting Tabular Functions

Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.
Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.

### Evaluating the Inverse of a Function, Given a Graph of the Original Function

We saw in Functions and Function Notation that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.

### Finding Inverses of Functions Represented by Formulas

Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula—for example,
   as a function of
    we can often find the inverse function by solving to obtain
    as a function of

### Finding Inverse Functions and Their Graphs

Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function
   restricted to the domain
   on which this function is one-to-one, and graph it as in .
Restricting the domain to
   makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.
We already know that the inverse of the toolkit quadratic function is the square root function, that is,
   What happens if we graph both
   and
   on the same set of axes, using the
   axis for the input to both

We notice a distinct relationship: The graph of
   is the graph of
   reflected about the diagonal line
   which we will call the identity line, shown in .
This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

### Key Concepts

1. If
   is the inverse of
   then
   See , , and .
2. Only some of the toolkit functions have an inverse. See .
3. For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
4. A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
5. For a tabular function, exchange the input and output rows to obtain the inverse. See .
6. The inverse of a function can be determined at specific points on its graph. See .
7. To find the inverse of a formula, solve the equation
   for
   as a function of
   Then exchange the labels
   and
   See , , and .
8. The graph of an inverse function is the reflection of the graph of the original function across the line
   See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find
   for each function.
For the following exercises, find a domain on which each function
   is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of
   restricted to that domain.
For the following exercises, use function composition to verify that
   and
   are inverse functions.

### Graphical

For the following exercises, use a graphing utility to determine whether each function is one-to-one.
For the following exercises, determine whether the graph represents a one-to-one function.
For the following exercises, use the graph of
   shown in .
For the following exercises, use the graph of the one-to-one function shown in .

### Numeric

For the following exercises, evaluate or solve, assuming that the function
   is one-to-one.
For the following exercises, use the values listed in  to evaluate or solve.

### Technology

For the following exercises, find the inverse function. Then, graph the function and its inverse.

### Real-World Applications

### Chapter Review Exercises

### Functions and Function Notation

For the following exercises, determine whether the relation is a function.
For the following exercises, evaluate

For the following exercises, determine whether the functions are one-to-one.
For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.
For the following exercises, graph the functions.
For the following exercises, use  to approximate the values.
For the following exercises, use the function
   to find the values in simplest form.

### Domain and Range

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

### Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

### Composition of Functions

For the following exercises, find
   and
    for each pair of functions.
For the following exercises, find
   and the domain for
   for each pair of functions.
For the following exercises, express each function
   as a composition of two functions
   and
   where

### Transformation of Functions

For the following exercises, sketch a graph of the given function.
For the following exercises, sketch the graph of the function
   if the graph of the function
   is shown in .
For the following exercises, write the equation for the standard function represented by each of the graphs below.
For the following exercises, determine whether each function below is even, odd, or neither.
For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

### Absolute Value Functions

For the following exercises, write an equation for the transformation of

For the following exercises, graph the absolute value function.

### Inverse Functions

For the following exercises, find
    for each function.
For the following exercise, find a domain on which the function
    is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of
    restricted to that domain.
For the following exercises, use a graphing utility to determine whether each function is one-to-one.

### Practice Test

For the following exercises, determine whether each of the following relations is a function.
For the following exercises, evaluate the function

  at the given input.
For the following exercises, use the functions
   to find the composite functions.
For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.
For the following exercises, determine whether the functions are even, odd, or neither.
For the following exercises, find the inverse of the function.
For the following exercises, use the graph of
   shown in .
For the following exercises, use the graph of the piecewise function shown in .
For the following exercises, use the values listed in .


# Linear Functions

## Introduction to Linear Functions

Imagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it may seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the fastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour. http://www.guinnessworldrecords.com/records-3000/fastest-growing-plant/ In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of change, such as the growth cycle of this bamboo plant, is a linear function.
Recall from Functions and Function Notation that a function is a relation that assigns to every element in the domain exactly one element in the range. Linear functions are a specific type of function that can be used to model many real-world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and how to relate them to data.


# Linear Functions

## Linear Functions

### Learning Objectives

1. Find the slope of a line (IA 3.2.1)
2. Find an equation of the line given two points (IA 3.3.3)

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

Linear functions are a specific type of function that can be used to model many real-world applications, such as the growth of a plant, earned salary, the distance a train travels over time, or the costs to start a new business. In this section, we will explore linear functions, their graphs, and how to find them using data points.

### Linear Function

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

where  is the initial or starting value of the function (when input, ), and  is the constant rate of change, or slope of the function. The  -intercept is at (  ).

When interpreting slope, 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.

### Practice Makes Perfect

Find the slope of the line.

### Objective 2: Find an equation of the line given two points. (IA 3.3.3)

### Find an Equation of the Line Given Two Points

When data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when two points are given by following the steps below.

### Practice Makes Perfect

Find an equation of the line given two points.
Just as with the growth of a bamboo plant, there are many situations that involve constant change over time. Consider, for example, the first commercial maglev train in the world, the Shanghai MagLev Train (). It carries passengers comfortably for a 30-kilometer trip from the airport to the subway station in only eight minuteshttp://www.chinahighlights.com/shanghai/transportation/maglev-train.htm.
Suppose a maglev train travels a long distance, and maintains a constant speed of 83 meters per second for a period of time once it is 250 meters from the station. How can we analyze the train’s distance from the station as a function of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the train’s distance from the station at a given point in time.

### Representing Linear Functions

The function describing the train’s motion is a linear function, which is defined as a function with a constant rate of change. This is a polynomial of degree 1. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. We will describe the train’s motion as a function using each method.

### Representing a Linear Function in Word Form

Let’s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.
1. The train’s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.
The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station.

### Representing a Linear Function in Function Notation

Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the slope-intercept form of a line, where
         is the input value,
         is the rate of change, and
         is the initial value of the dependent variable.
In the example of the train, we might use the notation
         where the total distance
         is a function of the time
  The rate,
         is 83 meters per second. The initial value of the dependent variable
         is the original distance from the station, 250 meters. We can write a generalized equation to represent the motion of the train.

### Representing a Linear Function in Tabular Form

A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in . From the table, we can see that the distance changes by 83 meters for every 1 second increase in time.

### Representing a Linear Function in Graphical Form

Another way to represent linear functions is visually, using a graph. We can use the function relationship from above,
  to draw a graph as represented in . Notice the graph is a line. When we plot a linear function, the graph is always a line.
The rate of change, which is constant, determines the slant, or slope of the line. The point at which the input value is zero is the vertical intercept, or , of the line. We can see from the graph that the y-intercept in the train example we just saw is
  and represents the distance of the train from the station when it began moving at a constant speed.
Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line

Ask yourself what numbers can be input to the function. In other words, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.

### Determining Whether a Linear Function Is Increasing, Decreasing, or Constant

The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an increasing function, as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in (a). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in (b). If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in (c).

### Interpreting Slope as a Rate of Change

In the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given input and output values. Recall that given two values for the input,
  and
  and two corresponding values for the output,
  and
  —which can be represented by a set of points,
  and
  —we can calculate the slope

Note that in function notation we can obtain two corresponding values for the output
  and
  for the function

  and
  so we could equivalently write
 indicates how the slope of the line between the points,
  and
  is calculated. Recall that the slope measures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is.

### Writing and Interpreting an Equation for a Linear Function

Recall from Equations and Inequalities that we wrote equations in both the slope-intercept form and the point-slope form. Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function
  in .
We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let’s choose
  and

Now we can substitute the slope and the coordinates of one of the points into the point-slope form.
If we want to rewrite the equation in the slope-intercept form, we would find
If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y-axis when the output value is 7. Therefore,
  We now have the initial value
  and the slope
  so we can substitute
  and
  into the slope-intercept form of a line.
So the function is
  and the linear equation would be

### Modeling Real-World Problems with Linear Functions

In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.

### Graphing Linear Functions

Now that we’ve seen and interpreted graphs of linear functions, let’s take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the y-intercept and slope. And the third method is by using transformations of the identity function

### Graphing a Function by Plotting Points

To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a grid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph. For example, given the function,
  we might use the input values 1 and 2. Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point
  Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point
  Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error.

### Graphing a Function Using y-intercept and Slope

Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its y-intercept, which is the point at which the input value is zero. To find the y-intercept, we can set
  in the equation.
The other characteristic of the linear function is its slope.
Let’s consider the following function.
The slope is
  Because the slope is positive, we know the graph will slant upward from left to right. The y-intercept is the point on the graph when
  The graph crosses the y-axis at
  Now we know the slope and the y-intercept. We can begin graphing by plotting the point
  We know that the slope is the change in the y-coordinate over the change in the x-coordinate. This is commonly referred to as rise over run,
  From our example, we have
  which means that the rise is 1 and the run is 2. So starting from our y-intercept
  we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in .

### Graphing a Function Using Transformations

Another option for graphing is to use a transformation of the identity function
  A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.

### Vertical Stretch or Compression

In the equation
  the
  is acting as the vertical stretch or compression of the identity function. When
  is negative, there is also a vertical reflection of the graph.  Notice in  that multiplying the equation of
  by
  stretches the graph of
  by a factor of
  units if
  and compresses the graph of
  by a factor of
  units if
  This means the larger the absolute value of
  the steeper the slope.

### Vertical Shift

In
  the
  acts as the vertical shift, moving the graph up and down without affecting the slope of the line. Notice in  that adding a value of
  to the equation of
  shifts the graph of
  a total of
  units up if
  is positive and
  units down if
  is negative.
Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.

### Writing the Equation for a Function from the Graph of a Line

Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at . We can see right away that the graph crosses the y-axis at the point
  so this is the y-intercept.
Then we can calculate the slope by finding the rise and run. We can choose any two points, but let’s look at the point
  To get from this point to the y-intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be
Substituting the slope and y-intercept into the slope-intercept form of a line gives

### Finding the x-intercept of a Line

So far we have been finding the y-intercepts of a function: the point at which the graph of the function crosses the y-axis. Recall that a function may also have an , which is the x-coordinate of the point where the graph of the function crosses the x-axis. In other words, it is the input value when the output value is zero.
To find the x-intercept, set a function
  equal to zero and solve for the value of
  For example, consider the function shown.
Set the function equal to 0 and solve for

The graph of the function crosses the x-axis at the point

### Describing Horizontal and Vertical Lines

There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line indicates a constant output, or y-value. In , we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use
  in the equation
  the equation simplifies to
  In other words, the value of the function is a constant. This graph represents the function

A vertical line indicates a constant input, or x-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.
A vertical line, such as the one in , has an x-intercept, but no y-intercept unless it’s the line
  This graph represents the line

### Determining Whether Lines are Parallel or Perpendicular

The two lines in  are parallel lines: they will never intersect. They have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the y-intercept. If we shifted one line vertically toward the other, they would become coincident.
We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the y-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.
Unlike parallel lines, perpendicular lines do intersect. Their intersection forms a right, or 90-degree, angle. The two lines in  are perpendicular.
Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and its reciprocal is
  So, if
  are negative reciprocals of one another, they can be multiplied together to yield

To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is
  and the reciprocal of
  is 8. To find the negative reciprocal, first find the reciprocal and then change the sign.
As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.
The product of the slopes is –1.

### Writing the Equation of a Line Parallel or Perpendicular to a Given Line

If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.

### Writing Equations of Parallel Lines

Suppose for example, we are given the equation shown.
We know that the slope of the line formed by the function is 3. We also know that the y-intercept is
  Any other line with a slope of 3 will be parallel to
   So the lines formed by all of the following functions will be parallel to

Suppose then we want to write the equation of a line that is parallel to
   and passes through the point
  This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of
   will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form.
So
   is parallel to
   and passes through the point

### Writing Equations of Perpendicular Lines

We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown.
The slope of the line is 2, and its negative reciprocal is
  Any function with a slope of
  will be perpendicular to
  So the lines formed by all of the following functions will be perpendicular to

As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to
  and passes through the point
  We already know that the slope is
  Now we can use the point to find the y-intercept by substituting the given values into the slope-intercept form of a line and solving for

The equation for the function with a slope of
  and a y-intercept of 2 is
So
  is perpendicular to
  and passes through the point
  Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature.

### Key Concepts

1. Linear functions can be represented in words, function notation, tabular form, and graphical form. See .
2. An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line. See .
3. Slope is a rate of change. The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line. See  and .
4. An equation for a linear function can be written from a graph. See .
5. The equation for a linear function can be written if the slope
  and initial value
  are known. See  and .
6. A linear function can be used to solve real-world problems given information in different forms. See , , and .
7. Linear functions can be graphed by plotting points or by using the y-intercept and slope. See  and .
8. Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See .
9. The equation for a linear function can be written by interpreting the graph. See .
10. The x-intercept is the point at which the graph of a linear function crosses the x-axis. See .
11. Horizontal lines are written in the form,
  See .
12. Vertical lines are written in the form,
  See .
13. Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See .
14. A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x- and y-values of the given point into the equation,
  and using the
  that results. Similarly, the point-slope form of an equation can also be used. See .
15. A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the equation of the curve can be written as a linear function.
For the following exercises, determine whether each function is increasing or decreasing.
For the following exercises, find the slope of the line that passes through the two given points.
For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither.
For the following exercises, find the x- and y-intercepts of each equation.
For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?
For the following exercises, write an equation for the line described.

### Graphical

For the following exercises, find the slope of the line graphed.
For the following exercises, write an equation for the line graphed.
For the following exercises, match the given linear equation with its graph in .
For the following exercises, sketch a line with the given features.
For the following exercises, sketch the graph of each equation.
For the following exercises, write the equation of the line shown in the graph.

### Numeric

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

### Technology

For the following exercises, use a calculator or graphing technology to complete the task.

### Extensions

For the following exercises, use the functions

### Real-World Applications


# Linear Functions

## Modeling with Linear Functions

### Learning Objectives

1. Graph and interpret applications of slope–intercept form of a linear function. (IA 3.2.5)

### Objective 1: Graph and interpret applications of slope–intercept form of a linear function. (IA 3.2.5)

### Graph and Interpret Applications of Slope–Intercept form of linear equations.

Many real-world applications are modeled by linear functions. We will take a look at a few applications here so you can see how equations written in slope–intercept form describe real world situations.
Usually, when a linear function uses real-world data, different letters are used to represent the variables, instead of using only  and  . The variables used remind us of what quantities are being measured.
Also, we often will need to adjust the axes in our rectangular coordinate system to different scales to accommodate the data in the application. Since many applications have both independent and dependent variables that are positive our graphs will lie primarily in Quadrant I.

### Linear Functions

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

where b is the initial or starting value of the function (when input, x=0), and m is the constant rate of change, or slope of the function. The y-intercept is at (0,b),

  .
When interpreting slope, 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.

### Practice Makes Perfect

Graph and interpret applications of slope–intercept form of a linear function.

Elan is a college student who plans to spend a summer in Seattle. Elan has saved $3,500 for their trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent this situation? What would be the x-intercept, and what can Elan learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.

### Building Linear Models from Verbal Descriptions

When building linear models to solve problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:
1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
2. Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
3. Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
4. Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
5. When needed, write a formula for the function.
6. Solve or evaluate the function using the formula.
7. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
8. Clearly convey your result using appropriate units, and answer in full sentences when necessary.
Now let’s take a look at the student in Seattle. In Elan’s situation, there are two changing quantities: time and money. The amount of money they have remaining while on vacation depends on how long they stay. We can use this information to define our variables, including units.
So, the amount of money remaining depends on the number of weeks:
  .
Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.
The rate of change is constant, so we can start with the linear model
  Then we can substitute the intercept and slope provided.
To find the t-intercept (horizontal axis intercept), we set the output to zero, and solve for the input.
The t-intercept (horizontal axis intercept) is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Elan will have no money left after 8.75 weeks.
When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before Elan saved $3,500, but the scenario discussed poses the question once they saved $3,500 because this is when the trip and subsequent spending starts. It is also likely that this model is not valid after the t-intercept (horizontal axis intercept), unless Elan uses a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is

In this example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.

### Using a Given Intercept to Build a Model

Some real-world problems provide the vertical axis intercept, which is the constant or initial value. Once the vertical axis intercept is known, the t-intercept (horizontal axis intercept) can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The y-intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.
Now we can set the function equal to 0, and solve for
  to find the x-intercept.
The x-intercept is the number of months it takes her to reach a balance of $0. The x-intercept is 4 months, so it will take Hannah four months to pay off her loan.

###  Using a Given Input and Output to Build a Model

Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.

### Using a Diagram to Build a Model

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.

### Modeling a Set of Data with Linear Functions

Real-world situations including two or more linear functions may be modeled with a system of linear equations. Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in .

### Key Concepts

1. We can use the same problem strategies that we would use for any type of function.
2. When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept. See .
3. Draw a diagram, where appropriate. See  and .
4. Check for reasonableness of the answer.
5. Linear models may be built by identifying or calculating the slope and using the y-intercept.

### Section Exercises

### Verbal

### Algebraic

For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped to 4,700. Assume this trend continues.
For the following exercises, consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.
For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.
For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.
For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted.

### Graphical

For the following exercises, use the graph in , which shows the profit,
  in thousands of dollars, of a company in a given year,
  where
  represents the number of years since 1980.
For the following exercises, use the graph in , which shows the profit,
  in thousands of dollars, of a company in a given year,
  where
  represents the number of years since 1980.

### Numeric

For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in . Assume that the house values are changing linearly.
For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in . Assume that the house values are changing linearly.

### Real-World Applications