# 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] .

#### Questions & Answers

the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard

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