# Functions

## Rates of Change and Behavior of Graphs

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

### 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

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, 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

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
  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 together. Vertical shifts are outside changes that affect the output (
  ) axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input (
  ) axis 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 right or left.

### 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. Functions whose graphs are symmetric about the y-axis are called even functions.
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, 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

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 investigate 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.

### 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

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,

### Solving an Absolute Value Inequality

Absolute value equations may not always involve equalities. Instead, we may need to solve an equation within a range of values. We would use an absolute value inequality to solve such an equation. An absolute value inequality is an equation of the form
where an expression
    (and possibly but not usually
  ) depends on a variable
  Solving the inequality means finding the set of all
     that satisfy the inequality. Usually this set will be an interval or the union of two intervals.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.
For example, we know that all numbers within 200 units of 0 may be expressed as
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of values

such that the distance between
    and 600 is less than 200. We represent the distance between

   and 600 as

This means our returns would be between $400 and $800.
Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and/or stretched or compressed absolute value function, where we must determine for which values of the input the function’s output will be negative or positive.

### Key Concepts

1. The absolute value function is commonly used to measure distances between points. See .
2. Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See .
3. The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See .
4. In an absolute value equation, an unknown variable is the input of an absolute value function.
5. If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See .
6. An absolute value equation may have one solution, two solutions, or no solutions. See .
7. An absolute value inequality is similar to an absolute value equation but takes the form
     It can be solved by determining the boundaries of the solution set and then testing which segments are in the set. See .
8. Absolute value inequalities can also be solved graphically. See .

### Section Exercise

### Verbal

### Algebraic

For the following exercises, solve the equations below and express the answer using set notation.
For the following exercises, find the x- and y-intercepts of the graphs of each function.
For the following exercises, solve each inequality and write the solution in interval notation.

### 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

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. Each of the toolkit functions has 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 the function at the indicated values:

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.

### 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.
For the following exercises, solve the absolute value equation.
For the following exercises, solve the inequality and express the solution using interval notation.

### 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

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 minutes.http://www.chinahighlights.com/shanghai/transportation/maglev-train.htm
Suppose a maglev train were to travel a long distance, and that the train 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, that 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 form known as 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

in which 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, 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 in  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, that is, 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).

### Calculating and Interpreting Slope

In the examples we have seen so far, we have had the slope provided for us. However, we often need to calculate the slope given input and output values. 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

as follows
where  is the vertical displacement and   is the horizontal displacement. Note in function notation 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. The greater the absolute value of the slope, the steeper the line is.

### Writing the Point-Slope Form of a Linear Equation

Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. Here, we will learn another way to write a linear function, the point-slope form.
The point-slope form is derived from the slope formula.
Keep in mind that the slope-intercept form and the point-slope form can be used to describe the same function. We can move from one form to another using basic algebra. For example, suppose we are given an equation in point-slope form,
. We can convert it to the slope-intercept form as shown.
Therefore, the same line can be described in slope-intercept form as

### Writing the Equation of a Line Using a Point and the Slope

The point-slope form is particularly useful if we know one point and the slope of a line. Suppose, for example, we are told that a line has a slope of 2 and passes through the point

We know that

and that

and
 We can substitute these values into the general point-slope equation.
If we wanted to then rewrite the equation in slope-intercept form, we apply algebraic techniques.
Both equations,

and
 describe the same line. See .

### Writing the Equation of a Line Using Two Points

The point-slope form of an equation is also useful if we know any two points through which a line passes. Suppose, for example, we know that a line passes through the points

and
 We can use the coordinates of the two points to find the slope.
Now we can use the slope we found and the coordinates of one of the points to find the equation for the line. Let use (0, 1) for our point.
As before, we can use algebra to rewrite the equation in the slope-intercept form.
Both equations describe the line shown in .

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

Now that we have written equations for linear functions in both the slope-intercept form and the point-slope form, we can choose which method to use 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
 We can use these points to calculate the slope.
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 wanted 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.

### Key Equations

### Key Concepts

1. The ordered pairs given by a linear function represent points on a line.
2. Linear functions can be represented in words, function notation, tabular form, and graphical form. See .
3. The rate of change of a linear function is also known as the slope.
4. An equation in the slope-intercept form of a line includes the slope and the initial value of the function.
5. The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point at which the line crosses the y-axis.
6. An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
7. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
8. A constant linear function results in a graph that is a horizontal line.
9. Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant. See .
10. 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 .
11. The slope and initial value can be determined given a graph or any two points on the line.
12. One type of function notation is the slope-intercept form of an equation.
13. The point-slope form is useful for finding a linear equation when given the slope of a line and one point. See .
14. The point-slope form is also convenient for finding a linear equation when given two points through which a line passes. See .
15. The equation for a linear function can be written if the slope  and initial value  are known. See , , and .
16. A linear function can be used to solve real-world problems. See  and .
17. A linear function can be written from tabular form. See .

### 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.

### Graphical

For the following exercises, find the slope of the lines graphed.
For the following exercises, write an equation for the lines graphed.

### 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

### Extensions

### Real-World Applications


# Linear Functions

## Graphs of Linear Functions

Two competing telephone companies offer different payment plans. The two plans charge the same rate per long distance minute, but charge a different monthly flat fee. A consumer wants to determine whether the two plans will ever cost the same amount for a given number of long distance minutes used. The total cost of each payment plan can be represented by a linear function. To solve the problem, we will need to compare the functions. In this section, we will consider methods of comparing functions using graphs.

### Graphing Linear Functions

In Linear Functions, we saw that that the graph of a linear function is a straight line. We were also able to see the points of the function as well as the initial value from a graph. By graphing two functions, then, we can more easily compare their characteristics.
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 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 , we can set  in the equation.
The other characteristic of the linear function is its slope  which is a measure of its steepness. Recall that the slope is the rate of change of the function. The slope of a function is equal to the ratio of the change in outputs to the change in inputs. Another way to think about the slope is by dividing the vertical difference, or rise, by the horizontal difference, or run. We encountered both the y-intercept and the slope in Linear Functions.
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 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 transformations 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

Recall that in Linear Functions, 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. A function may also have an -intercept, 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.
Notice that 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. Notice that 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 y-intercept of the other, they would become the same line.
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 1. So, if
   and
 are negative reciprocals of one another, they can be multiplied together to yield –1.
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 following equation.
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  We already know that the slope is 3. We just need to determine which value for  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 following function:
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.

### Solving a System of Linear Equations Using a Graph

A system of linear equations includes two or more linear equations. The graphs of two lines will intersect at a single point if they are not parallel. Two parallel lines can also intersect if they are coincident, which means they are the same line and they intersect at every point. For two lines that are not parallel, the single point of intersection will satisfy both equations and therefore represent the solution to the system.
To find this point when the equations are given as functions, we can solve for an input value so that  In other words, we can set the formulas for the lines equal to one another, and solve for the input that satisfies the equation.

### Key Concepts

1. Linear functions may be graphed by plotting points or by using the y-intercept and slope. See  and .
2. Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See .
3. The y-intercept and slope of a line may be used to write the equation of a line.
4. The x-intercept is the point at which the graph of a linear function crosses the x-axis. See  and .
5. Horizontal lines are written in the form,  See .
6. Vertical lines are written in the form,  See .
7. Parallel lines have the same slope.
8. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See .
9. 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 .
10. 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 .
11. A system of linear equations may be solved setting the two equations equal to one another and solving for  The y-value may be found by evaluating either one of the original equations using this x-value.
12. A system of linear equations may also be solved by finding the point of intersection on a graph. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
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?

### Graphical

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.
For the following exercises, find the point of intersection of each pair of lines if it exists. If it does not exist, indicate that there is no point of intersection.

### Extensions

For the following exercises, use the functions

### Real-World Applications


# Linear Functions

## Modeling with Linear Functions

Elan is a college student who plans to spend a summer in Seattle. Elan has saved $3,500 for the trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent the 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.

### Identifying Steps to Model and Solve Problems

When modeling scenarios with linear functions and solving 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.

### Building Linear Models

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.
1. Output:
 money remaining, in dollars
2. Input:
 time, in weeks
So, the amount of money remaining depends on the number of weeks:

We can also identify the initial value and the rate of change.
1. Initial Value: They saved $3,500, so $3,500 is the initial value for

2. Rate of Change: They anticipate spending $400 each week, so –$400 per week is the rate of change, or slope.
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 they are 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
 intercept, we set the output to zero, and solve for the input.
The
 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
 intercept, unless Elan will use a credit card and go into debt. The domain represents the set of input values, so the reasonable domain for this function is

In the above 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
 intercept, which is the constant or initial value. Once the
 intercept is known, the
 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
 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
 intercept.
The
  intercept is the number of months it takes her to reach a balance of $0. The
 -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 Model a Problem

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.

### Building Systems of Linear Models

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 .
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.
6. The x-intercept may be found by setting
  which is setting the expression
  equal to 0.
7. The point of intersection of a system of linear equations is the point where the x- and y-values are the same. See .
8. A graph of the system may be used to identify the points where one line falls below (or above) the other line.

### 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 afflicted.

### 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


# Linear Functions

## Fitting Linear Models to Data

A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relates the age of each student to the exam score received. In this section, we will examine one such diagram known as a scatter plot.

### Drawing and Interpreting Scatter Plots

A scatter plot is a graph of plotted points that may show a relationship between two sets of data. If the relationship is from a linear model, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear functions.  shows a sample scatter plot.
Notice this scatter plot does not indicate a linear relationship. The points do not appear to follow a trend. In other words, there does not appear to be a relationship between the age of the student and the score on the final exam.

### Finding the Line of Best Fit

Once we recognize a need for a linear function to model that data, the natural follow-up question is “what is that linear function?” One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can extend the line until we can verify the y-intercept. We can approximate the slope of the line by extending it until we can estimate the

### Recognizing Interpolation or Extrapolation

While the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the relationship will behave outside of the values for which we have data. We use a process known as interpolation when we predict a value inside the domain and range of the data. The process of extrapolation is used when we predict a value outside the domain and range of the data.
 compares the two processes for the cricket-chirp data addressed in . We can see that interpolation would occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44. Extrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or greater than 44.
There is a difference between making predictions inside the domain and range of values for which we have data and outside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model no longer applies after a certain point, it is sometimes called model breakdown. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when  that is in 50 years, the model would not apply because we could not account for factors fifty years in the future.

### Finding the Line of Best Fit Using a Graphing Utility

While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data valuesTechnically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values.. One such technique is called least squares regression and can be computed by many graphing calculators, spreadsheet software, statistical software, and many web-based calculatorsFor example, http://www.shodor.org/unchem/math/lls/leastsq.html. Least squares regression is one means to determine the line that best fits the data, and here we will refer to this method as linear regression.

### Distinguishing Between Linear and Non-Linear Models

As we saw above with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam scores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the correlation coefficient, which is a measure of how closely the line fits the data. Many graphing calculators require the user to turn a ”diagnostic on” selection to find the correlation coefficient, which mathematicians label as  The correlation coefficient provides an easy way to get an idea of how close to a line the data falls.
We should compute the correlation coefficient only for data that follows a linear pattern or to determine the degree to which a data set is linear. If the data exhibits a nonlinear pattern, the correlation coefficient for a linear regression is meaningless. To get a sense for the relationship between the value of  and the graph of the data,  shows some large data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the vertical axis shows the output.

### Predicting with a Regression Line

Once we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make predictions. As we learned above, a regression line is a line that is closest to the data in the scatter plot, which means that only one such line is a best fit for the data.

### Key Concepts

1. Scatter plots show the relationship between two sets of data. See .
2. Scatter plots may represent linear or non-linear models.
3. The line of best fit may be estimated or calculated, using a calculator or statistical software. See .
4. Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be used to predict values outside the domain and range of the data. See .
5. The correlation coefficient,  indicates the degree of linear relationship between data. See .
6. A regression line best fits the data. See .
7. The least squares regression line is found by minimizing the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related?

### Graphical

For the following exercises, match each scatterplot with one of the four specified correlations in  and .
For the following exercises, draw a best-fit line for the plotted data.

### Numeric

### Technology

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy.

### Extensions

For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span, (number of units sold, profit) for specific recorded years:
 .

### Real-World Applications

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific recorded years:

For the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year span, (number of units sold, profit) for specific recorded years:
 .
For the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the ten-year span (number of units sold, profit) for specific recorded years:

### Chapter Review Exercises

### Linear Functions

### Graphs of Linear Functions

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
For the following exercises, find the x- and y- intercepts of the given equation
For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

### Modeling with Linear Functions

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

For the following exercise, consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown to 2,500.
For the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly.
For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in . Assume that the house values are changing linearly.

### Fitting Linear Models to Data

For the following exercises, consider the data in , which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year.
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded years:

### Practice Test

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:
For the following exercises, use the graph in , showing the profit, , in thousands of dollars, of a company in a given year, , where  represents years since 1980.
For the following exercises, use , which shows the percent of unemployed persons 25 years or older who are college graduates in a particular city, by year.
For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:


# Polynomial and Rational Functions

## Introduction to Polynomial and Rational Functions

You don't need to dive very deep to feel the effects of pressure. As a person in their neighborhood pool moves eight, ten, twelve feet down, they often feel pain in their ears as a result of water and air pressure differentials. Pressure plays a much greater role at ocean diving depths.
Scuba and free divers are constantly negotiating the effects of pressure in order to experience enjoyable, safe, and productive dives. Gases in a person's respiratory system and diving apparatus interact according to certain physical properties, which upon discovery and evaluation are collectively known as the gas laws. Some are conceptually simple, such as the inverse relationship regarding pressure and volume, and others are more complex. While their formulas seem more straightforward than many you will encounter in this chapter, the gas laws are generally polynomial expressions.