# 5.7 Inverses and radical functions  (Page 5/7)

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## 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 function $\text{\hspace{0.17em}}C=\frac{20+0.4n}{100+n}\text{\hspace{0.17em}}$ represents the concentration $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ of an acid solution after $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ mL 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 for $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}C.\text{\hspace{0.17em}}$ Then 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 for $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}C.$

$\begin{array}{ccc}\hfill C& =& \frac{20+0.4n}{100+n}\hfill \\ \hfill C\left(100+n\right)& =& 20+0.4n\hfill \\ \hfill 100C+Cn& =& 20+0.4n\hfill \\ \hfill 100C-20& =& 0.4n-Cn\hfill \\ \hfill 100C-20& =& \left(0.4n-C\right)n\hfill \\ \hfill n& =& \frac{100C-20}{0.4-C}\hfill \end{array}$

Now evaluate this function at 35%, which is $\text{\hspace{0.17em}}C=0.35.$

$\begin{array}{ccc}\hfill n& =& \frac{100\left(0.35\right)-20}{0.4-0.35}\hfill \\ & =& \frac{15}{0.05}\hfill \\ & =& 300\hfill \end{array}$

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

Find the inverse of the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{x+3}{x-2}.$

${f}^{-1}\left(x\right)=\frac{2x+3}{x-1}$

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 $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ is the inverse of a function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is the inverse of the function $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$ See [link] .
• While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See [link] .
• To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See [link] and [link] .
• When finding the inverse of a radical function, we need a restriction on the domain of the answer. See [link] and [link] .
• Inverse and radical and functions can be used to solve application problems. See [link] and [link] .

## Verbal

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

It can be too difficult or impossible to solve for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}y.$

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.

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\sqrt{x}+4$

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\sqrt{x+3}-1$

$f\left(x\right)=3{x}^{2}+5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\infty ,0\right]$

${f}^{-1}\left(x\right)=-\sqrt{\frac{x-5}{3}}$

$f\left(x\right)=\sqrt{9-x}$

For the following exercises, find the inverse of the functions.

$f\left(x\right)={x}^{3}+5$

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\sqrt{x-5}$

$f\left(x\right)=3{x}^{3}+1$

$f\left(x\right)=4-{x}^{3}$

$\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\sqrt{4-x}$

$f\left(x\right)=4-2{x}^{3}$

For the following exercises, find the inverse of the functions.

$f\left(x\right)=\sqrt{2x+1}$

${f}^{-1}\left(x\right)=\frac{{x}^{2}-1}{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[0,\infty \right)$

$f\left(x\right)=\sqrt{3-4x}$

$f\left(x\right)=9+\sqrt{4x-4}$

$\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\frac{{\left(x-9\right)}^{2}+4}{4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[9,\infty \right)$

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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