3.7 Inverse functions (Page 5/9)

<< Chapter < Page

Algebra and trigonometry

Page 5 / 9

Chapter >> Page >

Try It

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

a. 3; b. 5.6

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Try It

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

$x = 3 y + 5$

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 $[2, \infty) .$

Got questions? Get instant answers now!

Try It

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

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

Got questions? Get instant answers now!

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] .

Got questions? Get instant answers now!

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.

John Reply

taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.

Faith

While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma

Kamara

which drugs can we use for ulcers

Ummi Reply

omeprazole

Kamara

what

Renee

what is this

Renee

is a drug

Kamara

of anti-ulcer

Kamara

Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit

Patrick

what is the function of lymphatic system

Nency Reply

Not really sure

Eli

to drain extracellular fluid all over the body.

asegid

The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of

asegid

to transport fluids fats proteins and lymphocytes to the blood stream as lymph

Adama

what is anatomy

Oyindarmola Reply

Anatomy is the identification and description of the structures of living things

Kamara

what's the difference between anatomy and physiology

Oyerinde Reply

Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.

AI-Robot

what is enzymes all about?

Mohammed Reply

Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems

Kamara

yes

Prince

how does the stomach protect itself from the damaging effects of HCl

Wulku Reply

little girl okay how does the stomach protect itself from the damaging effect of HCL

Wulku

it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL

Kamara

function of digestive system

Ali Reply

function of digestive

Ali

the diagram of the lungs

Adaeze Reply

what is the normal body temperature

Diya Reply

37 degrees selcius

Xolo

37°c

Stephanie

please why 37 degree selcius normal temperature

Mark

36.5

Simon

37°c

Iyogho

the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature

Stephanie

37A c

Wulku

what is anaemia

Diya Reply

anaemia is the decrease in RBC count hemoglobin count and PVC count

Eniola

what is the pH of the vagina

Diya Reply

how does Lysin attack pathogens

Diya

acid

Mary

I information on anatomy position and digestive system and there enzyme

Elisha Reply

anatomy of the female external genitalia

Muhammad Reply

Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)

Theophilus Reply

what's lochia albra

Kizito

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask

	7 Finding inverse functions and their graphs By OpenStax Read Online Course
	5 Evaluating the inverse of a function, given a graph of the By OpenStax Read Online Course
	11 Sociology 11 Race and Ethnicity MCQ By OpenStax Start Quiz
	16 AP 16 Neurological MCQ Exam By OpenStax Start Quiz
©flickr: Urko	Music Reviewer MCQ By Edgar Delgado Start Quiz
	Biology 302 Final By Dravida Mahadeo-J... Start Quiz
	12 AP 12 Nervous System Essay By OpenStax Start Flashcards
	4 AP 04 Tissue Level of Organization Essay By OpenStax Start Flashcards
	Introduction to sociology 2e By OpenStax Read Online Course
	4 Physiotherapy Flashcards Set 4 By Rhodes Start Flashcards
	Social Dances 2 By Marion Cabalfin Start Quiz
	College physics for ap® courses By OpenStax Read Online Course