Question #33262

1 Answer
Oct 30, 2016

Please see the explanation.

Explanation:

Given:

#f(x) = 2/(x + 1)#

And:

#g(f(x)) = 8/(x + 1)#

Find g(x):

Begin by finding #f^-1(x)#, because we know that:

#g(f(f^-1(x))) = g(x)#

Find #f^-1(x)#:

Substitute #f^-1(x)# for #x# in #f(x)#:

#f(f^-1(x)) = 2/(f^-1(x) + 1)#

By definition, substitute x for #f(f^-1(x))#:

#x = 2/(f^-1(x) + 1)#

Multiply both sides of the equation by #(f^-1(x) + 1)/x#

#f^-1(x) + 1 = 2/x#

Subtract 1 from both sides:

#f^-1(x) = 2/x - 1#

Substitute #2/x - 1# into #g(f(x))#:

#g(f(f^-1(x))) = 8/((2/x - 1) + 1)#

Remove the ()s in the denominator:

#g(f(f^-1(x))) = 8/(2/x - 1 + 1)#

-1 + 1 is zero:

#g(f(f^-1(x))) = 8/(2/x)#

Perform the division:

#g(f(f^-1(x))) = 8(x/2)#

Write the left side as #g(x)# and simplify the right:

#g(x) = 4x#