How do I evaluate inverse functions?

Question

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

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Answer 1 · 2017-11-12T08:09:21

#3^(-81)#

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)#.

Answer 2 · 2017-11-12T10:47:11

#g^(-1)(27) = 1#

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#