Question

Find the inverse of each of the functions below. (Each function is a separate question.)

1. f(x)=log base 4 of (x-5) + 3

2. g(x)= 3(1/2)^x-1 +6

Answer 1

\[f(x)=\log_{4} (x-5)+3\]

\[g(x)=3(\frac{ 1 }{ 2 })^{x-1}+6\]

Answer 2

Substitute y for x and x for f(x), then solve for y.
\[
f(x)=log_4(x-5)+3
\\\Rightarrow
x=log_4(y-5)+3
\\\Leftrightarrow
4^x=4^{log_4(y-5)+3}=4^3(y-5)
\\\Leftrightarrow
y=\frac{4^x}{4^3}+5=4^{x-3}+5
\]
So for the first question, we have \(f^{-1}(x)=4^{x-3}+5\)