<< Chapter < Page Chapter >> Page >
( f 1 f ) ( x ) = f 1 ( f ( x ) ) = f 1 ( y ) = x

This holds for all x in the domain of f . Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal , some functions do not have inverses.

Given a function f ( x ) , we can verify whether some other function g ( x ) is the inverse of f ( x ) by checking whether either g ( f ( x ) ) = x or f ( g ( x ) ) = x is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)

For example, y = 4 x and y = 1 4 x are inverse functions.

( f 1 f ) ( x ) = f 1 ( 4 x ) = 1 4 ( 4 x ) = x

and

( f f 1 ) ( x ) = f ( 1 4 x ) = 4 ( 1 4 x ) = x

A few coordinate pairs from the graph of the function y = 4 x are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function y = 1 4 x are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

Inverse function

For any one-to-one function     f ( x ) = y , a function f 1 ( x ) is an inverse function    of f if f 1 ( y ) = x . This can also be written as f 1 ( f ( x ) ) = x for all x in the domain of f . It also follows that f ( f 1 ( x ) ) = x for all x in the domain of f 1 if f 1 is the inverse of f .

The notation f 1 is read f inverse.” Like any other function, we can use any variable name as the input for f 1 , so we will often write f 1 ( x ) , which we read as f inverse of x . Keep in mind that

f 1 ( x ) 1 f ( x )

and not all functions have inverses.

Identifying an inverse function for a given input-output pair

If for a particular one-to-one function f ( 2 ) = 4 and f ( 5 ) = 12 , what are the corresponding input and output values for the inverse function?

The inverse function reverses the input and output quantities, so if

f ( 2 ) = 4 ,  then  f −1 ( 4 ) = 2 ; f ( 5 ) = 12 ,  then f −1 ( 12 ) = 5.

Alternatively, if we want to name the inverse function g , then g ( 4 ) = 2 and g ( 12 ) = 5.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Given that h 1 ( 6 ) = 2 , what are the corresponding input and output values of the original function h ?

h ( 2 ) = 6

Got questions? Get instant answers now!

Given two functions f ( x ) and g ( x ) , test whether the functions are inverses of each other.

  1. Determine whether f ( g ( x ) ) = x or g ( f ( x ) ) = x .
  2. If either statement is true, then both are true, and g = f 1 and f = g 1 . If either statement is false, then both are false, and g f 1 and f g 1 .

Testing inverse relationships algebraically

If f ( x ) = 1 x + 2 and g ( x ) = 1 x 2 , is g = f 1 ?

g ( f ( x ) ) = 1 ( 1 x + 2 ) 2 = x + 2 2 = x

so

g = f 1  and  f = g 1

This is enough to answer yes to the question, but we can also verify the other formula.

f ( g ( x ) ) = 1 1 x 2 + 2 = 1 1 x = x
Got questions? Get instant answers now!
Got questions? Get instant answers now!

If f ( x ) = x 3 4 and g ( x ) = x + 4 3 , is g = f 1 ?

Yes

Got questions? Get instant answers now!

Determining inverse relationships for power functions

If f ( x ) = x 3 (the cube function) and g ( x ) = 1 3 x , is g = f 1 ?

f ( g ( x ) ) = x 3 27 x

No, the functions are not inverses.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

If f ( x ) = ( x 1 ) 3 and g ( x ) = x 3 + 1 , is g = f 1 ?

Yes

Got questions? Get instant answers now!

Finding domain and range of inverse functions

The outputs of the function f are the inputs to f 1 , so the range of f is also the domain of f 1 . Likewise, because the inputs to f are the outputs of f 1 , the domain of f is the range of f 1 . We can visualize the situation as in [link] .

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 1

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask