Find inverse functions by solving f(x)=c for simple invertible functions and writing inverse expressions.
Find an inverse expression for a simple exponential function
Problem
Find the inverse of \(f(x)=2^x\). State the exponential's domain and range, swap \(x\) and \(y\), rewrite the equation in \(\text{base}-2\) logarithmic form, give an equivalent natural-log expression, state the inverse domain and range, and check \(f^{-1}(8)\).
Big Picture
What this problem is really about
The inverse of an exponential must recover an exponent, so logarithmic notation naturally appears after input and output are swapped. We’ll preserve the exponential’s positive range as the inverse domain, use a logarithm with the matching base, and verify the reversal with a familiar power.
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