# 3.7 Inverse functions  (Page 2/9)

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$\left({f}^{-1}\circ f\right)\left(x\right)={f}^{-1}\left(f\left(x\right)\right)={f}^{-1}\left(y\right)=x$

This holds for all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal , some functions do not have inverses.

Given a function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ we can verify whether some other function $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ is the inverse of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ by checking whether either $\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=x\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=x\text{\hspace{0.17em}}$ is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)

For example, $\text{\hspace{0.17em}}y=4x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=\frac{1}{4}x\text{\hspace{0.17em}}$ are inverse functions.

$\left({f}^{-1}\circ f\right)\left(x\right)={f}^{-1}\left(4x\right)=\frac{1}{4}\left(4x\right)=x$

and

$\left({f}^{}\circ {f}^{-1}\right)\left(x\right)=f\left(\frac{1}{4}x\right)=4\left(\frac{1}{4}x\right)=x$

A few coordinate pairs from the graph of the function $\text{\hspace{0.17em}}y=4x\text{\hspace{0.17em}}$ are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function $\text{\hspace{0.17em}}y=\frac{1}{4}x\text{\hspace{0.17em}}$ are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

## Inverse function

For any one-to-one function     $\text{\hspace{0.17em}}f\left(x\right)=y,\text{\hspace{0.17em}}$ a function $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)\text{\hspace{0.17em}}$ is an inverse function    of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}{f}^{-1}\left(y\right)=x.\text{\hspace{0.17em}}$ This can also be written as $\text{\hspace{0.17em}}{f}^{-1}\left(f\left(x\right)\right)=x\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ It also follows that $\text{\hspace{0.17em}}f\left({f}^{-1}\left(x\right)\right)=x\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the domain of $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ is the inverse of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$

The notation ${f}^{-1}$ is read $\text{“}f$ inverse.” Like any other function, we can use any variable name as the input for ${f}^{-1},$ so we will often write $\text{\hspace{0.17em}}{f}^{-1}\left(x\right),$ which we read as $“f$ inverse of $x.”$ Keep in mind that

${f}^{-1}\left(x\right)\ne \frac{1}{f\left(x\right)}$

and not all functions have inverses.

## Identifying an inverse function for a given input-output pair

If for a particular one-to-one function $\text{\hspace{0.17em}}f\left(2\right)=4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(5\right)=12,\text{\hspace{0.17em}}$ what are the corresponding input and output values for the inverse function?

The inverse function reverses the input and output quantities, so if

Alternatively, if we want to name the inverse function $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}g\left(4\right)=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(12\right)=5.$

Given that $\text{\hspace{0.17em}}{h}^{-1}\left(6\right)=2,\text{\hspace{0.17em}}$ what are the corresponding input and output values of the original function $\text{\hspace{0.17em}}h?\text{\hspace{0.17em}}$

$h\left(2\right)=6$

Given two functions $\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right),\text{\hspace{0.17em}}$ test whether the functions are inverses of each other.

1. Determine whether $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=x\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=x.$
2. If either statement is true, then both are true, and $\text{\hspace{0.17em}}g={f}^{-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f={g}^{-1}.\text{\hspace{0.17em}}$ If either statement is false, then both are false, and $\text{\hspace{0.17em}}g\ne {f}^{-1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\ne {g}^{-1}.$

## Testing inverse relationships algebraically

If $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x+2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{x}-2,\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g={f}^{-1}?$

$\begin{array}{ccc}\hfill g\left(f\left(x\right)\right)& =& \frac{1}{\left(\frac{1}{x+2}\right)}-2\hfill \\ & =& x+2-2\hfill \\ & =& x\hfill \end{array}$

so

This is enough to answer yes to the question, but we can also verify the other formula.

$\begin{array}{ccc}\hfill f\left(g\left(x\right)\right)& =& \frac{1}{\frac{1}{x}-2+2}\hfill \\ & =& \frac{1}{\frac{1}{x}}\hfill \\ & =& x\hfill \end{array}$

If $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt[\text{\hspace{0.17em}}3]{x+4},\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g={f}^{-1}?$

Yes

## Determining inverse relationships for power functions

If $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ (the cube function) and $\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{3}x,\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g={f}^{-1}?$

$f\left(g\left(x\right)\right)=\frac{{x}^{3}}{27}\ne x$

No, the functions are not inverses.

If $\text{\hspace{0.17em}}f\left(x\right)={\left(x-1\right)}^{3}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}g\left(x\right)=\sqrt[3]{x}+1,\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g={f}^{-1}?$

Yes

## Finding domain and range of inverse functions

The outputs of the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ are the inputs to $\text{\hspace{0.17em}}{f}^{-1},\text{\hspace{0.17em}}$ so the range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is also the domain of $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$ Likewise, because the inputs to $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ are the outputs of $\text{\hspace{0.17em}}{f}^{-1},\text{\hspace{0.17em}}$ the domain of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is the range of $\text{\hspace{0.17em}}{f}^{-1}.\text{\hspace{0.17em}}$ We can visualize the situation as in [link] .

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
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Amit
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Dorbor
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Biswajit
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Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
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siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
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Nikki
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Rubben
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Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
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Okey tell me, what's your problem is?
Navin