# 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] .

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this