Polynomial and Rational Functions

# Inverses and Radical Functions

### Learning Objectives

In this section, you will:

• Find the inverse of an invertible polynomial function.
• Restrict the domain to find the inverse of a polynomial function.

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 volumein terms of the radiusHowever, 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 forin 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 functionsandare inverse functions if for every coordinate pair inthere 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. Only one-to-one functions have inverses. Recall that a one-to-one function has a unique output value for each input value and passes the horizontal line test.

For example, suppose a water runoff collector is built in the shape of a parabolic trough as shown in (Figure). 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, withmeasured horizontally andmeasured vertically, with the origin at the vertex of the parabola. See (Figure).

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 formOur 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 depththe width will be given byso we need to solve the equation above forand 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 positivevalues. On this domain, we can find an inverse by solving for the input variable:

This is not a function as written. We are limiting ourselves to positivevalues, so we eliminate the negative solution, giving us the inverse function we’re looking for.

Becauseis the distance from the center of the parabola to either side, the entire width of the water at the top will beThe 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 functionThis use of “–1” is reserved to denote inverse functions. To denote the reciprocal of a functionwe would need to write

An important relationship between inverse functions is that they “undo” each other. Ifis the inverse of a function
thenis the inverse of the functionIn other words, whatever the functiondoes to
undoes it—and vice-versa.

and

Note that the inverse switches the domain and range of the original function.

### Verifying Two Functions Are Inverses of One Another

Two functions,
andare inverses of one another if for allin the domain of
and

### How To

Given a polynomial function, find the inverse of the function by restricting the domain in such a way that the new function is one-to-one.

1. Replacewith
2. Interchangeand
3. Solve forand rename the function

### Verifying Inverse Functions

Show thatandare inverses, for.

We must show thatand

Therefore,
andare inverses.

Show that
andare inverses.

and

### Finding the Inverse of a Cubic Function

Find the inverse of the function

This is a transformation of the basic cubic toolkit function, and based on our knowledge of that function, we know it is one-to-one. Solving for the inverse by solving for

#### Analysis

Look at the graph ofandNotice that one graph is the reflection of the other about the lineThis is always the case when graphing a function and its inverse function.

Also, since the method involved interchangingandnotice corresponding points. Ifis on the graph ofthenis on the graph ofSinceis on the graph ofthenis on the graph ofSimilarly, sinceis on the graph ofthenis on the graph ofSee (Figure).

### Try It

Find the inverse function of

### 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.

### Restricting the Domain

If a function is not one-to-one, it cannot have an inverse. If we restrict the domain of the function so that it becomes one-to-one, thus creating a new function, this new function will have an inverse.

### How To

Given a polynomial function, restrict the domain of a function that is not one-to-one and then find the inverse.

1. Restrict the domain by determining a domain on which the original function is one-to-one.
2. Replacewith
3. Interchangeand
4. Solve forand rename the function or pair of function
5. Revise the formula forby ensuring that the outputs of the inverse function correspond to the restricted domain of the original function.

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

Find the inverse function of

The original functionis not one-to-one, but the function is restricted to a domain oforon which it is one-to-one. See (Figure).

To find the inverse, start by replacingwith the simple variable

This is not a function as written. We need to examine the restrictions on the domain of the original function to determine the inverse. Since we reversed the roles ofandfor the originalwe looked at the domain: the values
could assume. When we reversed the roles ofand
this gave us the valuescould assume. For this function,so for the inverse, we should havewhich is what our inverse function gives.

1. The domain of the original function was restricted toso the outputs of the inverse need to be the same,and we must use the + case:
2. The domain of the original function was restricted toso the outputs of the inverse need to be the same,and we must use the – case:

#### Analysis

On the graphs in (Figure), we see the original function graphed on the same set of axes as its inverse function. Notice that together the graphs show symmetry about the lineThe coordinate pairis on the graph ofand the coordinate pairis on the graph ofFor any coordinate pair, ifis on the graph ofthenis on the graph ofFinally, observe that the graph ofintersects the graph ofon the linePoints of intersection for the graphs ofandwill always lie on the line

### Finding the Inverse of a Quadratic Function When the Restriction Is Not Specified

Restrict the domain and then find the inverse of

We can see this is a parabola with vertex atthat opens upward. Because the graph will be decreasing on one side of the vertex and increasing on the other side, we can restrict this function to a domain on which it will be one-to-one by limiting the domain to

To find the inverse, we will use the vertex form of the quadratic. We start by replacingwith a simple variable,then solve for

Now we need to determine which case to use. Because we restricted our original function to a domain ofthe outputs of the inverse should be the same, telling us to utilize the + case

If the quadratic had not been given in vertex form, rewriting it into vertex form would be the first step. This way we may easily observe the coordinates of the vertex to help us restrict the domain.[/hidden-answer]

#### Analysis

Notice that we arbitrarily decided to restrict the domain onWe could just have easily opted to restrict the domain onin which caseObserve the original function graphed on the same set of axes as its inverse function in (Figure). Notice that both graphs show symmetry about the lineThe coordinate pairis on the graph ofand the coordinate pairis on the graph ofObserve from the graph of both functions on the same set of axes that

and

Finally, observe that the graph ofintersects the graph ofalong the line

### Try It

Find the inverse of the functionon the domain

#### 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.

### How To

Given a radical function, find the inverse.

1. Determine the range of the original function.
2. Replace withthen solve for
3. If necessary, restrict the domain of the inverse function to the range of the original function.

### Finding the Inverse of a Radical Function

Restrict the domain of the functionand then find the inverse.

Note that the original function has rangeReplacewiththen solve for

Recall that the domain of this function must be limited to the range of the original function.

#### Analysis

Notice in (Figure) that the inverse is a reflection of the original function over the lineBecause the original function has only positive outputs, the inverse function has only positive inputs.

### Try It

Restrict the domain and then find the inverse of the function

#### Solving Applications of Radical Functions

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.

### Solving an Application with a Cubic Function

A mound of gravel is in the shape of a cone with the height equal to twice the radius. The volume of the cone in terms of the radius is given by

Find the inverse of the functionthat determines the volumeof a cone and is a function of the radiusThen use the inverse function to calculate the radius of such a mound of gravel measuring 100 cubic feet. Use

Start with the given function forNotice that the meaningful domain for the function issince negative radii would not make sense in this context nor would a radius of 0. Also note the range of the function (hence, the domain of the inverse function) isSolve forin terms ofusing the method outlined previously. Note that in real-world applications, we do not swap the variables when finding inverses. Instead, we change which variable is considered to be the independent variable.

This is the result stated in the section opener. Now evaluate this forand

#### 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 the Domain of a Radical Function Composed with a Rational Function

Find the domain of the function

Because a square root is only defined when the quantity under the radical is non-negative, we need to determine whereThe output of a rational function can change signs (change from positive to negative or vice versa) at x-intercepts and at vertical asymptotes. For this equation, the graph could change signs at

To determine the intervals on which the rational expression is positive, we could test some values in the expression or sketch a graph. While both approaches work equally well, for this example we will use a graph as shown in (Figure).

This function has two x-intercepts, both of which exhibit linear behavior near the x-intercepts. There is one vertical asymptote, corresponding to a linear factor; this behavior is similar to the basic reciprocal toolkit function, and there is no horizontal asymptote because the degree of the numerator is larger than the degree of the denominator. There is a y-intercept at

From the y-intercept and x-intercept atwe can sketch the left side of the graph. From the behavior at the asymptote, we can sketch the right side of the graph.

From the graph, we can now tell on which intervals the outputs will be non-negative, so that we can be sure that the original functionwill be defined.has domainor in interval notation,

#### 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.

### Finding the Inverse of a Rational Function

The functionrepresents the concentrationof an acid solution aftermL of 40% solution has been added to 100 mL of a 20% solution. First, find the inverse of the function; that is, find an expression forin terms ofThen use your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.

We first want the inverse of the function in order to determine how many mL we need for a given concentration. We will solve forin terms of

Now evaluate this function at 35%, which is

We can conclude that 300 mL of the 40% solution should be added.

### Try It

Find the inverse of the function

Access these online resources for additional instruction and practice with inverses and radical functions.

### Key Concepts

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

### Section Exercises

#### Verbal

Explain why we cannot find inverse functions for all polynomial functions.

It can be too difficult or impossible to solve forin terms of

Why must we restrict the domain of a quadratic function when finding its inverse?

When finding the inverse of a radical function, what restriction will we need to make?

We will need a restriction on the domain of the answer.

The inverse of a quadratic function will always take what form?

#### 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 withpositive real numbers.

#### Real-World Applications

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

An object dropped from a height of 200 meters has a height,in meters afterseconds have lapsed, such thatExpressas a function of height,and find the time to reach a height of 50 meters.

5.53 seconds

An object dropped from a height of 600 feet has a height,in feet afterseconds have elapsed, such thatExpress
as a function of heightand find the time to reach a height of 400 feet.

The volume,of a sphere in terms of its radius,is given byExpressas a function ofand find the radius of a sphere with volume of 200 cubic feet.

3.63 feet

The surface area,of a sphere in terms of its radius,is given byExpressas a function ofand find the radius of a sphere with a surface area of 1000 square inches.

A container holds 100 mL of a solution that is 25 mL acid. IfmL of a solution that is 60% acid is added, the functiongives the concentration,as a function of the number of mL added,Expressas a function ofand determine the number of mL that need to be added to have a solution that is 50% acid.

250 mL

The periodin seconds, of a simple pendulum as a function of its lengthin feet, is given by . Expressas a function ofand determine the length of a pendulum with period of 2 seconds.

The volume of a cylinder ,in terms of radius,and height,is given byIf a cylinder has a height of 6 meters, express the radius as a function ofand find the radius of a cylinder with volume of 300 cubic meters.

3.99 meters

The surface area,of a cylinder in terms of its radius,and height,is given byIf the height of the cylinder is 4 feet, express the radius as a function ofand find the radius if the surface area is 200 square feet.

The volume of a right circular cone,in terms of its radius,and its height,is given byExpressin terms ofif the height of the cone is 12 feet and find the radius of a cone with volume of 50 cubic inches.

1.99 inches