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

 Page 5 / 7

## 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.

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

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

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

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

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

what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions
the polar co-ordinate of the point (-1, -1)
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
tanh`(x-iy) =A+iB, find A and B
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
Ajith
exponential series
Naveen
yeah
Morosi
prime number?
Morosi
what is subgroup
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1