Question

Question #33262

Answer 1

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`