3.7 Inverse functions (Page 5/9)

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Using the graph in [link] , (a) find $g^{- 1} (1),$ and (b) estimate $g^{- 1} (4) .$

a. 3; b. 5.6

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Finding inverses of functions represented by formulas

Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula—for example, $y$ as a function of $x —$ we can often find the inverse function by solving to obtain $x$ as a function of $y .$

How To

Given a function represented by a formula, find the inverse.

Make sure $f$ is a one-to-one function.
Solve for $x .$
Interchange $x$ and $y .$

Inverting the fahrenheit-to-celsius function

Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.

C = \frac{5}{9} (F - 32)

\begin{matrix} C & = & \frac{5}{9} (F - 32) \\ C \cdot \frac{9}{5} & = & F - 32 \\ F & = & \frac{9}{5} C + 32 \end{matrix}

By solving in general, we have uncovered the inverse function. If

C = h (F) = \frac{5}{9} (F - 32),

then

F = h^{- 1} (C) = \frac{9}{5} C + 32

In this case, we introduced a function $h$ to represent the conversion because the input and output variables are descriptive, and writing $C^{- 1}$ could get confusing.

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Solve for $x$ in terms of $y$ given $y = \frac{1}{3} (x - 5) .$

$x = 3 y + 5$

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Solving to find an inverse function

Find the inverse of the function $f (x) = \frac{2}{x - 3} + 4.$

\begin{matrix} y & = & \frac{2}{x - 3} + 4 & Set up an equation . \\ y - 4 & = & \frac{2}{x - 3} & Subtract 4 from both sides . \\ x - 3 & = & \frac{2}{y - 4} & Multiply both sides by x - 3 and divide by y - 4. \\ x & = & \frac{2}{y - 4} + 3 & Add 3 to both sides . \end{matrix}

So $f^{- 1} (y) = \frac{2}{y - 4} + 3$ or $f^{- 1} (x) = \frac{2}{x - 4} + 3.$

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Solving to find an inverse with radicals

Find the inverse of the function $f (x) = 2 + \sqrt{x - 4} .$

\begin{matrix} y & = & 2 + \sqrt{x - 4} \\ {(y - 2)}^{2} & = & x - 4 \\ x & = & {(y - 2)}^{2} + 4 \end{matrix}

So $f^{- 1} (x) = {(x - 2)}^{2} + 4.$

The domain of $f$ is $[4, \infty) .$ Notice that the range of $f$ is $[2, \infty),$ so this means that the domain of the inverse function $f^{- 1}$ is also $[2, \infty) .$

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What is the inverse of the function $f (x) = 2 - \sqrt{x} ?$ State the domains of both the function and the inverse function.

$f^{- 1} (x) = {(2 - x)}^{2}; domain of f : [0, \infty); domain of f^{- 1} : (- \infty, 2]$

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Finding inverse functions and their graphs

Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function $f (x) = x^{2}$ restricted to the domain $[0, \infty),$ on which this function is one-to-one, and graph it as in [link] .

Graph of f(x). — Quadratic function with domain restricted to [0, ∞).

Restricting the domain to $[0, \infty)$ makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.

We already know that the inverse of the toolkit quadratic function is the square root function, that is, $f^{- 1} (x) = \sqrt{x} .$ What happens if we graph both $f$ and $f^{- 1}$ on the same set of axes, using the $x -$ axis for the input to both $f and f^{- 1} ?$

We notice a distinct relationship: The graph of $f^{- 1} (x)$ is the graph of $f (x)$ reflected about the diagonal line $y = x,$ which we will call the identity line, shown in [link] .

Graph of f(x) and f^(-1)(x). — Square and square-root functions on the non-negative domain

This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

Finding the inverse of a function using reflection about the identity line

Given the graph of $f (x)$ in [link] , sketch a graph of $f^{- 1} (x) .$

This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of $(0, \infty)$ and range of $(- \infty, \infty),$ so the inverse will have a domain of $(- \infty, \infty)$ and range of $(0, \infty) .$

If we reflect this graph over the line $y = x,$ the point $(1, 0)$ reflects to $(0, 1)$ and the point $(4, 2)$ reflects to $(2, 4) .$ Sketching the inverse on the same axes as the original graph gives [link] .

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Questions & Answers

If c is the cost function for a particular product, find the marginal cost functions and their values at x=10 a. c(x) = 800+ 0.04x + 0.0002x² b. c(x) = 250 + 100x + 0.001x²

Mamush Reply

how can I find set theory

Ephraim Reply

how can I find set theory

Jarvis

is there an error on the one about the dime's thickness? says 2.2x10⁶=0.00135 m

Patrick Reply

hi, interested in algebra

Makan Reply

how to reduce an equation?

Makan

by manipulation of both side

9(y+8)-27 is 9y+45. Why can't you reduce that to y+5? I know that's wrong but can't explain why

Patrick Reply

when you reduce an equation to its simplest terms, you can't change the value of the equation. reducing it to y + 5 is equivalent to dividing it by 9 which changes the value. you can multiply it by 1 or 9/9 which would give 9(y + 5). multiplying it by one does not change the value.

Philip

Given a polynomial expression, factor out the greatest common factor.

Hanu Reply

WHAT IS QUADRATIC EQUATION?

Charles Reply

WHAT IS SYSTEM OF LINEAR INEWUALITIES?

Charles

WHAT IS SYSTEM OF LINEAR INEWUALITIES?

Charles

complex perform

Angel

what is equation?

Charles Reply

what are equations?

Charles

Definition of economics according to karl Marx Thomas malthus Jeremy bentham David Ricardo J.K

Rakiya

Please help me is assignment

Rakiya

The 47th problem of Euclid

Kenneth

show that the set of all natural number form semi group under the composition of addition

Nikhil Reply

what is the meaning

Dominic

explain and give four Example hyperbolic function

Lukman Reply

_3_2_1

felecia

⅗ ⅔½

felecia

_½+⅔-¾

felecia

The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu

SABAL Reply

1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3

Pawel

2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31

Pawel

Ifeanyi

on number 2 question How did you got 2x +2

Ifeanyi

combine like terms. x + x + 2 is same as 2x + 2

Pawel

x*x=2

felecia

2+2x=

felecia

×/×+9+6/1

Debbie

Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22

Naagmenkoma

Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?

mariel Reply

Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113

Pawel

how do I set up the problem?

Harshika Reply

what is a solution set?

Harshika

find the subring of gaussian integers?

Rofiqul

hello, I am happy to help!

Shirley Reply

please can go further on polynomials quadratic

Abdullahi

hi mam

Mark

I need quadratic equation link to Alpa Beta

Abdullahi Reply

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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