First, I need to find f –1(x), g –1(x), and ( f o g)–1(x):
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Inverting f (x):
Inverting g(x):
Finding the composed function: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved
Inverting the composed function:
Now I'll compose the inverses of f(x) and g(x) to find the formula for (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 (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)
(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)
( 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.
