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Chapter 3: Problem 51

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=2^{x-5} $$

Short Answer

Expert verified
The inverse function of \(f(x)=2^{x-5}\) is \(f^{-1}(x) = \log_2{x} + 5\).

Step by step solution

01

Replace function notation with 'y'

Replace \(f(x)\) with \(y\) in the given function: $$ y = 2^{x-5} $$
02

Swap 'x' and 'y'

Switch \(x\) and \(y\) in the equation: $$ x = 2^{y-5} $$
03

Solve for 'y'

Our goal is to isolate \(y\) on one side of the equation. To do that, we'll first take the logarithm (base 2) of both sides: $$ \log_2{x} = \log_2{2^{y-5}} $$ The equation simplifies to: $$ \log_2{x} = y - 5 $$ Now, we'll add 5 to both sides to completely isolate \(y\): $$ y = \log_2{x} + 5 $$ After finding y, replace 'y' with \(f^{-1}(x)\), which represents the inverse function of \(f(x)\): $$ f^{-1}(x) = \log_2{x} + 5 $$ So, the inverse function of \(f(x) = 2^{x-5}\) is \(f^{-1}(x) = \log_2{x} + 5\).

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Most popular questions from this chapter

Show that $$ \sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y $$ for all real numbers \(x\) and \(y\).

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