# Functions

## Inverse Functions


### Learning Objectives

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

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

A composite function is a two-step function and can have numerical or variable inputs.
  is read as “f of g of x”.
To evaluate a composite function, we always start by evaluating the inner function and then evaluate the outer function in terms of the inner function.

### Practice Makes Perfect


Find and evaluate composite functions.

For each of the following function pairs find:

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

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

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


### Practice Makes Perfect


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

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

###  Verifying That Two Functions Are Inverse Functions

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

### Finding Domain and Range of Inverse Functions

The outputs of the function 
   are the inputs to 
   so the range of 
   is also the domain of 
   Likewise, because the inputs to 
   are the outputs of 
   the domain of 
   is the range of 
   We can visualize the situation as in .
When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of 
   is 
   because a square “undoes” a square root; but the square is only the inverse of the square root on the domain 
   since that is the range of 
  
  
We can look at this problem from the other side, starting with the square (toolkit quadratic) function 
   If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and –3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the “inverse” is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.
In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function 
   with its domain limited to 
   which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).
If 
   on 
   then the inverse function is 
  
  
1. The domain of 
   = range of 
   = 
  
  
2. The domain of 
   = range of 
   = 
  
  

###  Finding and Evaluating Inverse Functions

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

### Inverting Tabular Functions

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

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

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

### Finding Inverses of Functions Represented by Formulas

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

### Finding Inverse Functions and Their Graphs

Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function 
   restricted to the domain 
   on which this function is one-to-one, and graph it as in .
Restricting the domain to 
   makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.
We already know that the inverse of the toolkit quadratic function is the square root function, that is, 
   What happens if we graph both 
   and 
   on the same set of axes, using the 
   axis for the input to both 
  
  
We notice a distinct relationship: The graph of 
   is the graph of 
   reflected about the diagonal line 
   which we will call the identity line, shown in .
This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

### Key Concepts

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

### Section Exercises


### Verbal


### Algebraic

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

### Graphical

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

### Numeric

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

### Technology

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

### Real-World Applications


### Chapter Review Exercises


### Functions and Function Notation

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

### Domain and Range

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

### Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from 
  
  
For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

### Composition of Functions

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

### Transformation of Functions

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

### Absolute Value Functions

For the following exercises, write an equation for the transformation of 
  
  
For the following exercises, graph the absolute value function.

### Inverse Functions

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

### Practice Test

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