How do I evaluate inverse functions?

Let f(x)=3^x and g(x)=f(f(x)). Evaluate g^-1(27).

2 Answers
Nov 12, 2017

3^(-81)

Explanation:

If f(x) = 3^x, and g(x) = f(f(x)), then first we must evaluate g(x).

g(x) = f(3^x)
g(x) = 3^(3x)

An inverse function works like this: If f(x) = x, then f^(-1)(x) = 1/x or x^(-1).
Here, g(x) = 3^(3x) so g^(-1)(x) = 1/3^(3x)

Now g^(-1)(27) must be evaluated. Input it into the above:
g^(-1)(27) = 1/3^(3*27)
g^(-1)(27) = 1/3^81 or 3^(-81).

Nov 12, 2017

g^(-1)(27) = 1

Explanation:

Given:

f(x) = 3^x

g(x) = f(f(x)) = 3^(3^x)

If g(x) = 27 then:

3^color(blue)(3^x) = 27 = 3^color(blue)(3)

As a real valued function f(x) = 3^x is one to one.

So we can deduce that the exponents are equal:

3^x = 3 = 3^1

and hence:

x = 1

So:

g^(-1)(27) = 1