ARGH! I typed in the exact same answer as the one in the "I need help" paragraph but it won't register?

It won't work because the fraction is not simplified in 30/4. You have to simplify it into 15/2 and then it will work

Is there ever a circumstance where f(g(x)) = x but g(f(x)) isn't equal to x?

Interesting question! This could occur if f(x) fails to be 1-to-1 (and so fails to be invertible). For example, if f(x) = x^2 and g(x) = sqrt(x), then f(g(x)) = [sqrt(x)]^2 = x but g(f(x)) = sqrt(x^2) = |x|. Have a blessed, wonderful day!

I NEED HELP! I'm working on a problem and have the g(f(x))=2(x+1/2)-1 as a problem I'm just trying to figure it out with the fraction. i stink writing stuff out so if you can understand please help me;(

Your working is wrong bokeum. It will be 2x+2/2 which wil become 2x/2 +2/2 to simplify it. It will become x+1-1 the (-1) from the original question. so it will equal x.

in 2, how did they get from =4(4/1 x−10)+10 to =x-40+10

Multiply everything in brackets by 4. 4*x/4 = x 4*-10 = -40 4(x/4 - 10) + 10 = x-40+10. You stated the problem in the wrong form. It should be 4(x/4 - 10) + 10 instead of 4(4/x - 10) +10

if i take 2 functions f(x) and g(x) and composite of only one of them results in x , can they be said as inverse functions ?

For 𝑓 and 𝑔 to be inverse functions, the domain of either function must be equal to the range of the other function. If 𝑓(𝑎) = 𝑏, but 𝑔(𝑏) ≠ 𝑎, then 𝑓 maps 𝑎 to 𝑏, but 𝑔 does not map 𝑏 to 𝑎. This means that the range of 𝑔 is not equal to the domain of 𝑓, and thereby 𝑓 and 𝑔 are not inverse functions.

I have a conjecture. Given two functions f and g, f and g are inverses of each other if and only if f and g are invertible and f(g(x)) = x. Can someone help me prove or disprove it?

To test if 2 functions are inverses, you need to do the test in both directions: f(g(x)) = x AND g(f(x)) = x

Main content

Precalculus

Course: Precalculus > Unit 1

Lesson 5: Verifying inverse functions by composition

Verifying inverse functions by composition

CCSS.Math:

HSF.BF.B.4

HSF.BF.B.4b

Google Classroom

Learn how to verify whether two functions are inverses by composing them. For example, are f(x)=5x-7 and g(x)=x/5+7 inverse functions?

This article includes a lot of function composition. If you need a review on this subject, we recommend that you go here before reading this article.

Inverse functions, in the most general sense, are functions that "reverse" each other. For example, if a function takes a to b, then the inverse must take b to a.

Let's take functions f and g for example: f, left parenthesis, x, right parenthesis, equals, start fraction, x, plus, 1, divided by, 3, end fraction and g, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1.

Notice how f, left parenthesis, 5, right parenthesis, equals, 2 and g, left parenthesis, 2, right parenthesis, equals, 5.

[Please show me the calculations]

Here we see that when we apply f followed by g, we get the original input back. Written as a composition, this is g, left parenthesis, f, left parenthesis, 5, right parenthesis, right parenthesis, equals, 5.

But for two functions to be inverses, we have to show that this happens for all possible inputs regardless of the order in which f and g are applied. This gives rise to the inverse composition rule.

The inverse composition rule

These are the conditions for two functions f and g to be inverses:

f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals, x for all x in the domain of g
g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals, x for all x in the domain of f

This is because if f and g are inverses, composing f and g (in either order) creates the function that for every input returns that input. We call this function “the identity function".

Example 1: Functions f and g are inverses

Let's use the inverse composition rule to verify that f and g above are indeed inverse functions.

Recall that f, left parenthesis, x, right parenthesis, equals, start fraction, x, plus, 1, divided by, 3, end fraction and g, left parenthesis, x, right parenthesis, equals, 3, x, minus, 1.

Let's find f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis and g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis.

f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis	g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis
$\begin{aligned} f(\greenD{g(x)})&=\dfrac{\greenD{g(x)}+1}{3}\\\\&=\dfrac{\greenD{3x-1}+1}{3}\\\\&=\dfrac{3x}{3}\\\\&=x\\\end{aligned}$	$\qquad\qquad \begin{aligned}g(\purpleC{f(x)})&=3\left(\purpleC{f(x)}\right)-1\\\\&=3\left(\purpleC{\dfrac{x+1}{3}}\right)-1\\\\&=x+1-1\\\\&=x\\\end{aligned}$

So we see that functions f and g are inverses because f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals, x and g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals, x.

Example 2: Functions f and g are not inverses

If f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis or g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis is not equal to x, then f and g cannot be inverses.

Let's try this for f, left parenthesis, x, right parenthesis, equals, 5, x, minus, 7 and g, left parenthesis, x, right parenthesis, equals, start fraction, x, divided by, 5, end fraction, plus, 7.

f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis	g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis
$\begin{aligned} f(\greenD{g(x)})&=5(\greenD{g(x)})-7\\\\&=5\left(\greenD{\dfrac{x}{5}+7}\right)-7\\\\&=x+35-7\\\\&=x+28\end{aligned}\qquad$	$\qquad\begin{aligned} g(\purpleC{f(x)})&=\dfrac{\purpleC{f(x)}}{5}+7\\\\&=\dfrac{\purpleC{5x-7}}{5}+7\\\\&=x-\dfrac75+7\\\\&=x+\dfrac{28}{5}\\\end{aligned}$

So functions f and g are not inverses because f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, does not equal, x and g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, does not equal, x.

(Note here, that we could have concluded that f and g were not inverses after showing that f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals, x, plus, 28.)

Check your understanding

In general, to check if f and g are inverse functions, we can compose them. If the result is x, the functions are inverses. Otherwise, they are not.

1) f, left parenthesis, x, right parenthesis, equals, 2, x, plus, 7 and h, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 7, divided by, 2, end fraction

Write simplified expressions for f, left parenthesis, h, left parenthesis, x, right parenthesis, right parenthesis and h, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis in terms of x.

f, left parenthesis, h, left parenthesis, x, right parenthesis, right parenthesis, equals

h, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals

Are functions f and h inverses?

[I need help!]

2) f, left parenthesis, x, right parenthesis, equals, 4, x, plus, 10 and g, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, 4, end fraction, x, minus, 10

Write simplified expressions for f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis and g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis in terms of x.

f, left parenthesis, g, left parenthesis, x, right parenthesis, right parenthesis, equals

g, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals

Are functions f and g inverses?

[I need help!]

3) f, left parenthesis, x, right parenthesis, equals, start fraction, 2, divided by, 3, end fraction, x, minus, 8 and h, left parenthesis, x, right parenthesis, equals, start fraction, 3, divided by, 2, end fraction, left parenthesis, x, plus, 8, right parenthesis

f, left parenthesis, h, left parenthesis, x, right parenthesis, right parenthesis, equals

h, left parenthesis, f, left parenthesis, x, right parenthesis, right parenthesis, equals

Are functions f and h inverses?

[I need help!]

Want to join the conversation?

Sort by:

YanSu
Posted 7 years ago. Direct link to YanSu's post “ARGH! I typed in the exac...”
ARGH! I typed in the exact same answer as the one in the "I need help" paragraph but it won't register?
Button navigates to signup pageComment on YanSu's post “ARGH! I typed in the exac...”
(20 votes)
Answer
- brand3636
  Posted 7 years ago. Direct link to brand3636's post “It won't work because the...”
  It won't work because the fraction is not simplified in 30/4. You have to simplify it into 15/2 and then it will work
  Comment on brand3636's post “It won't work because the...”
  (34 votes)
alexander erlichherzog
Posted 5 years ago. Direct link to alexander erlichherzog's post “Is there ever a circumsta...”
Is there ever a circumstance where f(g(x)) = x but g(f(x)) isn't equal to x?
Button navigates to signup pageButton navigates to signup page
(18 votes)
Answer
- Ian Pulizzotto
  Posted 5 years ago. Direct link to Ian Pulizzotto's post “Interesting question! Th...”
  Interesting question!
  
  This could occur if f(x) fails to be 1-to-1 (and so fails to be invertible).
  For example, if f(x) = x^2 and g(x) = sqrt(x), then f(g(x)) = [sqrt(x)]^2 = x but g(f(x)) = sqrt(x^2) = |x|.
  
  Have a blessed, wonderful day!
  Comment on Ian Pulizzotto's post “Interesting question! Th...”
  (36 votes)
Marc LaVincci
Posted 7 years ago. Direct link to Marc LaVincci's post “What happens if both simp...”
What happens if both simplify down to x+3 or any other values?
Are they still inverse of each other?
Button navigates to signup pageComment on Marc LaVincci's post “What happens if both simp...”
(6 votes)
Answer
business.morris.edits
Posted 5 days ago. Direct link to business.morris.edits's post “i just love math and calc...”
i just love math and calculus
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
ashley franklin
Posted 7 years ago. Direct link to ashley franklin's post “I NEED HELP! I'm working ...”
I NEED HELP! I'm working on a problem and have the g(f(x))=2(x+1/2)-1 as a problem I'm just trying to figure it out with the fraction. i stink writing stuff out so if you can understand please help me;(
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Meerah Tahir
  Posted 5 years ago. Direct link to Meerah Tahir's post “Your working is wrong bok...”
  Your working is wrong bokeum. It will be 2x+2/2 which wil become 2x/2 +2/2 to simplify it. It will become x+1-1 the (-1) from the original question. so it will equal x.
  Button navigates to signup page
  (2 votes)
Lily Martin
Posted 7 years ago. Direct link to Lily Martin's post “in 2, how did they get fr...”
in 2, how did they get from =4(4/1 x−10)+10 to =x-40+10
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Edison Elder
  Posted 6 years ago. Direct link to Edison Elder's post “Multiply everything in br...”
  Multiply everything in brackets by 4. 4*x/4 = x 4*-10 = -40
  4(x/4 - 10) + 10 = x-40+10. You stated the problem in the wrong form.
  It should be 4(x/4 - 10) + 10 instead of 4(4/x - 10) +10
  Comment on Edison Elder's post “Multiply everything in br...”
  (2 votes)
J R Adhwaith
Posted 3 years ago. Direct link to J R Adhwaith's post “if i take 2 functions f(x...”
if i take 2 functions f(x) and g(x) and composite of only one of them results in x , can they be said as inverse functions ?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Jerry Nilsson
  Posted 3 years ago. Direct link to Jerry Nilsson's post “For 𝑓 and 𝑔 to be inver...”
  For 𝑓 and 𝑔 to be inverse functions, the domain of either function must be equal to the range of the other function.
  
  If 𝑓(𝑎) = 𝑏, but 𝑔(𝑏) ≠ 𝑎,
  then 𝑓 maps 𝑎 to 𝑏, but 𝑔 does not map 𝑏 to 𝑎.
  This means that the range of 𝑔 is not equal to the domain of 𝑓,
  and thereby 𝑓 and 𝑔 are not inverse functions.
  Button navigates to signup page
  (3 votes)
kea241199
Posted 2 years ago. Direct link to kea241199's post “I have a conjecture. Give...”
I have a conjecture. Given two functions f and g, f and g are inverses of each other if and only if f and g are invertible and f(g(x)) = x. Can someone help me prove or disprove it?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kim Seidel
  Posted 2 years ago. Direct link to Kim Seidel's post “To test if 2 functions ar...”
  To test if 2 functions are inverses, you need to do the test in both directions:
  f(g(x)) = x AND g(f(x)) = x
  Comment on Kim Seidel's post “To test if 2 functions ar...”
  (2 votes)
Akashitakeru110
Posted 5 years ago. Direct link to Akashitakeru110's post “how do I verify thatf^-1(...”
how do I verify thatf^-1(x)=f(x)
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
Zhonghuang He
Posted 6 years ago. Direct link to Zhonghuang He's post “The inverse function mean...”
The inverse function mean that the f(g(x)) multiply g(f(x)) is equal x^2, right? For example, f(x)=2x, g(x)=x/2. f(g(x)) multiply g(f(x)) is x^2.
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Julia Pockat
  Posted 6 years ago. Direct link to Julia Pockat's post “No, an inverse function i...”
  No, an inverse function is a function that undoes the affect of an equation. If a coordinate point of one function is (0,4), its inverse is (4,0). So in your case, you have f(x) is the inverse of g(x), and y=2x. In order to undo this and find the inverse, you can switch the x and the y values, and solve for y. 2y=x, and dividing both sides by two, you get x/2. g(x) would be equal to x/2. Does this make sense?
  Comment on Julia Pockat's post “No, an inverse function i...”
  (1 vote)

Precalculus

Course: Precalculus > Unit 1

The inverse composition rule

Example 1: Functions ffff and gggg are inverses

Example 2: Functions ffff and gggg are not inverses

Check your understanding

1) f(x)=2x+7f(x)=2x+7f(x)=2x+7f, left parenthesis, x, right parenthesis, equals, 2, x, plus, 7 and h(x)=x−72h(x)=\dfrac{x-7}{2}h(x)=2x−7​h, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 7, divided by, 2, end fraction

2) f(x)=4x+10f(x)=4x+10f(x)=4x+10f, left parenthesis, x, right parenthesis, equals, 4, x, plus, 10 and g(x)=14x−10g(x)=\dfrac{1}{4}x-10g(x)=41​x−10g, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, 4, end fraction, x, minus, 10

Want to join the conversation?

Example 1: Functions f and g are inverses

Example 2: Functions f and g are not inverses

1) f, left parenthesis, x, right parenthesis, equals, 2, x, plus, 7 and h, left parenthesis, x, right parenthesis, equals, start fraction, x, minus, 7, divided by, 2, end fraction

2) f, left parenthesis, x, right parenthesis, equals, 4, x, plus, 10 and g, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, 4, end fraction, x, minus, 10