Question

5-36. use multiple representations to create the inverse of $y = 2^x$. is the inverse a function? hint:

Explanation:

Step1: Swap x and y variables

Swap \(x\) and \(y\) in the original function:
\(x = 2^y\)

Step2: Solve for y using logarithms

Rewrite the exponential equation in logarithmic form (since \(a^b = c\) is equivalent to \(\log_a c = b\)):
\(y = \log_2 x\)

Step3: Check if inverse is a function

Use the vertical line test: any vertical line drawn on the graph of \(y = \log_2 x\) intersects the graph at most once. Alternatively, the original function \(y=2^x\) is one-to-one (passes horizontal line test), so its inverse is a function.

Answer:

The inverse of \(y=2^x\) is \(y = \log_2 x\), and this inverse is a function.