Demo MathHelp.com
Join MathHelp.com
Login

Standardized Test Prep
- ACCUPLACER Math
- ACT Math
- ALEKS Math
- ASVAB Math
- CBEST Math
- CLEP Math
- FTCE Math
- GED Math
- GMAT Math
- GRE Math
- HESI Math
- Math Placement Test
- NES Math
- PERT Math
- PRAXIS Math
- SAT Math
- TEAS Math
- TSI Math
- VPT Math
- + more tests
K12 Math
- 5th Grade Math
- 6th Grade Math
- Pre-Algebra
- Algebra 1
- Geometry
- Algebra 2
College Math
- College Pre-Algebra
- Introductory Algebra
- Intermediate Algebra
- College Algebra
Homeschool Math
- Pre-Algebra
- Algebra 1
- Geometry
- Algebra 2

Function Composition, Function Inverses, and How They Relate

Sets of Points Functions at Points Functions into Functions Domains & Decompositions Word Probs

Purplemath

One use of function composition is for checking if two functions are inverses of each other. If you compose the two functions and end up with just x, then the functions are inverses of each other. The lesson on inverse functions explains and demonstrates how this works.

Content Continues Below

MathHelp.com

However, there is another connection between function composition and function inversion. You may be expected to show (or to "discover") this connection by doing some symbolic computations.

Given f(x) = 2x − 1 and , find the following:

f⁻¹(x)
g⁻¹(x)
(f ∘ g)⁻¹(x)
(g⁻¹ ∘ f⁻¹)(x)

What can you conclude?

This involves a lot of steps, so I'll stop talking and just show you how it goes.

First, I need to find f⁻¹(x), g⁻¹(x), and (f ∘ g)(x). Then I can find (f ∘ g)⁻¹(x) and g⁻¹(x) ∘ f⁻¹(x).

Inverting f (x):

Affiliate

f (x) = 2x − 1

y = 2x − 1

y + 1 = 2x

(y + 1)/2 = x (x + 1)/2 = y (x + 1)/2 = f ⁻¹(x)

Inverting g(x):

g(x) = (1/2)x + 4 y = (1/2)x + 4 y − 4 = (1/2)x 2(y − 4) = x 2y − 8 = x 2x − 8 = y 2x − 8 = g⁻¹(x)

Finding the composed function:

(f ∘ g)(x) = f (g(x)) = f ((1/2)x + 4) = 2((1/2)x + 4) − 1 = x + 8 − 1 = x + 7

Inverting the composed function:

(f ∘ g)(x) = x + 7

y = x + 7

y − 7 = x

x − 7 = y

x − 7 = (f ∘ g)⁻¹(x)

Content Continues Below

Now I'll compose the inverses of f (x) and g(x) to find the formula for (g⁻¹ ∘ f⁻¹)(x):

(g⁻¹ ∘ f⁻¹)(x) = g⁻¹(f⁻¹(x)) = g⁻¹( (x + 1)/2 ) = 2( (x + 1)/2 ) − 8 = (x + 1) − 8 = x − 7 = (g⁻¹ ∘ f⁻¹)(x)

Note that the inverse of the composition (( f ∘ g)⁻¹(x)) gives the same result as does the composition of the inverses ((g⁻¹ ∘ f⁻¹)(x)). So I would conclude that:

(f ∘ g)⁻¹(x) = (g⁻¹ ∘ f⁻¹)(x)

Having completed this exercise, I notice that the order of f and g is reversed. What if I'd composed the inverses in the other order?

(f⁻¹ ∘ g⁻¹)(x) = f⁻¹(g⁻¹(x)= f⁻¹(2x − 8)= [(2x − 8) + 1]/2= (2x − 7)/2

This is clearly not equal to the inverse of the composition f ∘ g, so clearly the order of the inverted functions in the composition *must* be opposite the order of the composed functions in the inverse.

What is the relationship between the inverse of a composition and the composition of the inverses?

When you have two invertible functions, the inverse of the composition of these functions is equal to the composition of the inverses of the functions, but in the reverse order. In other words, given f (x), g(x), and their composition (f ∘ g)(x), all invertible, then:

(f ∘ g)⁻¹(x) = g⁻¹(x) ∘ f⁻¹(x)

I'll say it again: The order of the functions is reversed in the composition of the inverses from the order they'd been in the inverse of the composition. Don't forget this detail!

Affiliate

While it is beyond the scope of this lesson to prove the above equality, I can tell you that this equality is indeed always true, assuming that the inverses and compositions exist — that is, assuming there aren't any problems with the domains and ranges and such, you can invert the composition by composing the inverses, but in the reverse order.

URL: https://www.purplemath.com/modules/fcncomp6.htm

Page 1 Page 2 Page 3 Page 4 Page 5

Select a Course Below

Standardized Test Prep

ACCUPLACER Math
ACT Math
ALEKS Math
ASVAB Math
CBEST Math
CLEP Math
FTCE Math
GED Math
GMAT Math
GRE Math
HESI Math
Math Placement Test
NES Math
PERT Math
PRAXIS Math
SAT Math
TEAS Math
TSI Math
VPT Math
+ more tests

Terms of Use
Privacy / Cookies
Contact

About Purplemath
About the Author
Tutoring from PM

Advertising
Linking to PM
Site licencing

Select a Course Below

Function Composition, Function Inverses, and How They Relate

Purplemath

MathHelp.com

What is the relationship between the inverse of a composition and the composition of the inverses?

Select a Course Below

Standardized Test Prep

K12 Math

College Math

Homeschool Math

Share This Page