# Polynomial and Rational Functions

## Inverses and Radical Functions

Park rangers and other trail managers may construct rock piles, stacks, or other arrangements, usually called cairns, to mark trails or other landmarks. (Rangers and environmental scientists discourage hikers from doing the same, in order to avoid confusion and preserve the habitats of plants and animals.) A cairn in the form of a mound of gravel is in the shape of a cone with the height equal to twice the radius.
The volume is found using a formula from elementary geometry.
We have written the volume 
  in terms of the radius 
  However, in some cases, we may start out with the volume and want to find the radius. For example: A customer purchases 100 cubic feet of gravel to construct a cone shape mound with a height twice the radius. What are the radius and height of the new cone? To answer this question, we use the formula
This function is the inverse of the formula for 
  in terms of 
  
In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process.

### Finding the Inverse of a Polynomial Function

Two functions 
  and 
  are inverse functions if for every coordinate pair in 
  there exists a corresponding coordinate pair in the inverse function, 
  In other words, the coordinate pairs of the inverse functions have the input and output interchanged.
For a function to have an inverse, it must be one-to-one.
For example, suppose the Sustainability Club builds a water runoff collector in the shape of a parabolic trough as shown in . We can use the information in the figure to find the surface area of the water in the trough as a function of the depth of the water.
Because it will be helpful to have an equation for the parabolic cross-sectional shape, we will impose a coordinate system at the cross section, with 
  measured horizontally and 
  measured vertically, with the origin at the vertex of the parabola. See .
From this we find an equation for the parabolic shape. We placed the origin at the vertex of the parabola, so we know the equation will have form 
  Our equation will need to pass through the point (6, 18), from which we can solve for the stretch factor 
 

Our parabolic cross section has the equation
We are interested in the surface area of the water, so we must determine the width at the top of the water as a function of the water depth. For any depth 
  the width will be given by 
  so we need to solve the equation above for 
  and find the inverse function. However, notice that the original function is not one-to-one, and indeed, given any output there are two inputs that produce the same output, one positive and one negative.
To find an inverse, we can restrict our original function to a limited domain on which it is one-to-one. In this case, it makes sense to restrict ourselves to positive 
  values. On this domain, we can find an inverse by solving for the input variable:
This is not a function as written. Since we are limiting ourselves to positive 
  values in the original function, we can eliminate the negative solution, which gives us the inverse function we’re looking for.
Because 
  is the distance from the center of the parabola to either side, the entire width of the water at the top will be 
  The trough is 3 feet (36 inches) long, so the surface area will then be:
This example illustrates two important points:
1. When finding the inverse of a quadratic, we have to limit ourselves to a domain on which the function is one-to-one.
2. The inverse of a quadratic function is a square root function. Both are toolkit functions and different types of power functions.
Functions involving roots are often called radical functions. While it is not possible to find an inverse of most polynomial functions, some basic polynomials do have inverses. Such functions are called invertible functions, and we use the notation 
 

Warning: 
  is not the same as the reciprocal of the function 
  This use of “–1” is reserved to denote inverse functions. To denote the reciprocal of a function 
  we would need to write 
 

An important relationship between inverse functions is that they “undo” each other. If 
  is the inverse of a function 
 
then 
  is the inverse of the function 
  In other words, whatever the function 
  does to 
 
 
  undoes it—and vice-versa. More formally, we write
and

### Restricting the Domain to Find the Inverse of a Polynomial Function

So far, we have been able to find the inverse functions of cubic functions without having to restrict their domains. However, as we know, not all cubic polynomials are one-to-one. Some functions that are not one-to-one may have their domain restricted so that they are one-to-one, but only over that domain. The function over the restricted domain would then have an inverse function. Since quadratic functions are not one-to-one, we must restrict their domain in order to find their inverses.

### Solving Applications of Radical Functions

Notice that the functions from previous examples were all polynomials, and their inverses were radical functions. If we want to find the inverse of a radical function, we will need to restrict the domain of the answer because the range of the original function is limited.

### 

Radical functions are common in physical models, as we saw in the section opener. We now have enough tools to be able to solve the problem posed at the start of the section.

### Determining the Domain of a Radical Function Composed with Other Functions

When radical functions are composed with other functions, determining domain can become more complicated.

### Finding Inverses of Rational Functions

As with finding inverses of quadratic functions, it is sometimes desirable to find the inverse of a rational function, particularly of rational functions that are the ratio of linear functions, such as in concentration applications.

### Key Concepts

1. The inverse of a quadratic function is a square root function.
2. If 
 
is the inverse of a function 
 
then 
 
is the inverse of the function 
 
See .
3. While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See .
4. To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See  and .
5. When finding the inverse of a radical function, we need a restriction on the domain of the answer. See  and .
6. Inverse and radical and functions can be used to solve application problems. See  and .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the inverse of the function on the given domain.
For the following exercises, find the inverse of the functions.
For the following exercises, find the inverse of the functions.

### Graphical

For the following exercises, find the inverse of the function and graph both the function and its inverse.
For the following exercises, use a graph to help determine the domain of the functions.

### Technology

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with y-coordinates given.

###  Extensions

For the following exercises, find the inverse of the functions with 
  positive real numbers.

### Real-World Applications

For the following exercises, determine the function described and then use it to answer the question.