QUESTION 8 KZNS16 Given \( p(x)=3^{x} \). 3.1 Write down the equation of \( p^{-1} \), the inverse of \( p \), in the form \( y=\ldots \) 3.2 Sketch in your ANSWER BOOK the graphs of \( p \) and \( p^{-1} \) on the same system of axes. Show clearly all the intercepts with the axes and at least one other point on cach graph. Determine the values of \( x \) for which \( p^{-1}(x) \leq 3 \)

To find the inverse function of \( p(x) = 3^x \), we start by replacing \( p(x) \) with \( y \): 1. Replace \( p(x) \) with \( y \): \[ y = 3^x \] 2. Solve for \( x \): \[ x = \log_3(y) \] 3. Thus, the inverse function is: \[ p^{-1}(y) = \log_3(y) \quad \text{(or in the traditional form, } y = \log_3(x) \text{)} \] Now for the graph, plot \( p(x) \) which is an exponential function and increases rapidly as \( x \) increases, and \( p^{-1}(x) \) which is a logarithmic function that increases slowly. For \( p^{-1}(x) \leq 3 \): 1. Set up the inequality: \[ \log_3(x) \leq 3 \] 2. Solve for \( x \): \[ x \leq 3^3 = 27 \] Thus, the values of \( x \) for which \( p^{-1}(x) \leq 3 \) are all \( x \) such that \( x \leq 27 \).