The lesson on inverse
functions explains
how to use function 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?

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):

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Inverting f
(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)

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)

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
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.