# 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.

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

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\sqrt[3]{x-5}$

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

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

$\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\sqrt[3]{4-x}$

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

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

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

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

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

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

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

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