# 1.7 Inverse functions  (Page 4/10)

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The domain of function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(1,\infty \right)\text{\hspace{0.17em}}$ and the range of function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(\mathrm{-\infty },-2\right).\text{\hspace{0.17em}}$ Find the domain and range of the inverse function.

The domain of function $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(-\infty \text{,}-2\right)\text{\hspace{0.17em}}$ and the range of function $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(1,\infty \right).$

## Finding and evaluating inverse functions

Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.

## Inverting tabular functions

Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.

Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.

## Interpreting the inverse of a tabular function

A function $\text{\hspace{0.17em}}f\left(t\right)\text{\hspace{0.17em}}$ is given in [link] , showing distance in miles that a car has traveled in $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ minutes. Find and interpret $\text{\hspace{0.17em}}{f}^{-1}\left(70\right).$

 30 50 70 90 20 40 60 70

The inverse function takes an output of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and returns an input for $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ So in the expression $\text{\hspace{0.17em}}{f}^{-1}\left(70\right),\text{\hspace{0.17em}}$ 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ 90 minutes, so $\text{\hspace{0.17em}}{f}^{-1}\left(70\right)=90.\text{\hspace{0.17em}}$ The interpretation of this is that, to drive 70 miles, it took 90 minutes.

Alternatively, recall that the definition of the inverse was that if $\text{\hspace{0.17em}}f\left(a\right)=b,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}{f}^{-1}\left(b\right)=a.\text{\hspace{0.17em}}$ By this definition, if we are given $\text{\hspace{0.17em}}{f}^{-1}\left(70\right)=a,\text{\hspace{0.17em}}$ then we are looking for a value $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ so that $\text{\hspace{0.17em}}f\left(a\right)=70.\text{\hspace{0.17em}}$ In this case, we are looking for a $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ so that $\text{\hspace{0.17em}}f\left(t\right)=70,\text{\hspace{0.17em}}$ which is when $\text{\hspace{0.17em}}t=90.$

Using [link] , find and interpret (a) and (b)

 30 50 60 70 90 20 40 50 60 70
1. $f\left(60\right)=50.\text{\hspace{0.17em}}$ In 60 minutes, 50 miles are traveled.
2. ${f}^{-1}\left(60\right)=70.\text{\hspace{0.17em}}$ To travel 60 miles, it will take 70 minutes.

## Evaluating the inverse of a function, given a graph of the original function

We saw in Functions and Function Notation that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.

Given the graph of a function, evaluate its inverse at specific points.

1. Find the desired input on the y -axis of the given graph.
2. Read the inverse function’s output from the x -axis of the given graph.

## Evaluating a function and its inverse from a graph at specific points

A function $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ is given in [link] . Find $\text{\hspace{0.17em}}g\left(3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{g}^{-1}\left(3\right).$

To evaluate $g\left(3\right),\text{\hspace{0.17em}}$ we find 3 on the x -axis and find the corresponding output value on the y -axis. The point $\text{\hspace{0.17em}}\left(3,1\right)\text{\hspace{0.17em}}$ tells us that $\text{\hspace{0.17em}}g\left(3\right)=1.$

To evaluate $\text{\hspace{0.17em}}{g}^{-1}\left(3\right),\text{\hspace{0.17em}}$ recall that by definition $\text{\hspace{0.17em}}{g}^{-1}\left(3\right)\text{\hspace{0.17em}}$ means the value of x for which $\text{\hspace{0.17em}}g\left(x\right)=3.\text{\hspace{0.17em}}$ By looking for the output value 3 on the vertical axis, we find the point $\text{\hspace{0.17em}}\left(5,3\right)\text{\hspace{0.17em}}$ on the graph, which means $\text{\hspace{0.17em}}g\left(5\right)=3,\text{\hspace{0.17em}}$ so by definition, $\text{\hspace{0.17em}}{g}^{-1}\left(3\right)=5.\text{\hspace{0.17em}}$ See [link] .

what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil