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Composition of Functions:
     Inverse Functions and Composition (page 5 of 5)

Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition

The lesson on inverse functions covers using composition to verify that two functions are inverses of each other. However, there is another connection between composition and inversion:

• Given f (x) = 2x – 1 and g(x) = (¹/₂)x + 4,
  find f ^–1(x), g ^–1(x), ( f o g)^–1(x), and  (g^–1 o f ^–1)(x).
  
  What can you conclude?   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

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 o g)^–1(x):

f (x) = 2x – 1
    y = 2x – 1
    y + 1 = 2x
    ^{(y + 1)}/₂ = x
    ^{(x + 1)}/₂ = y
    ^{(x + 1)}/₂ =  f ^–1(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)

( f o g)(x) = f (g(x)) = f ((¹/₂)x + 4)
    = 2((¹/₂)x + 4) – 1
    = x + 8 – 1
    = x + 7

( f o g)(x) = x + 7
    y = x + 7
    y – 7 = x
    x – 7 = y
    x – 7 = ( f o g)^–1(x)

Now I'll use these inverses to evaluate (g^–1 o f ^–1)(x):

(g^–1 o 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 o f ^–1)(x)

Note that the inverse of the composition (( f o g)^–1(x)) gives the same result as does the composition of the inverses ((g^–1 o f ^–1)(x)). So I would conclude that

( f o g)^–1(x) = (g^–1 o f ^–1)(x)

While it is beyond the scope of this lesson to prove this equality, I can tell you that this equality is always true, assuming that the inverses and compositions exist -- that is, assuming you don't have any problems with the domains and ranges and such.

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