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

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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