Question

In Exercises \(25-32,\) use parametric graphing to graph \(f, f^{-1},\) and \(y=x\) $$f(x)=3^{x}$$

Short answer

The graph of the function \(f(x) = 3^x\) will be an upward-curving line. The graph of its inverse \(f^{-1}(x) = log_3(x)\) is a line reflecting \(f(x) = 3^x\) at the line \(y = x\). The line y = x is a straight line passing through the origin with a gradient of 1.

Step-by-step solution

Graph the function

To make a graph, we need values for x and their corresponding y values which are calculated by using the function f(x). Let's choose some values for x like -2, -1, 0, 1, 2 and calculate the corresponding y values which will give us: (-2, 0.11), (-1, 0.33), (0, 1), (1, 3), (2, 9). Plot these points and connect them to get the graph.

Find the inverse of the function

The inverse of the function \(f(x) = 3^x\) is obtained by swapping the x and y coordinates and solving for the new y. This becomes \(x = 3^y\), and by converting to logarithmic form, we find the inverse function \(f^{-1}(x) = log_3(x)\). By substituting values for x (we can use the earlier chosen x values), we can calculate the y values. The points to plot for the inverse function will be (0.11, -2), (0.33, -1), (1, 0), (3, 1), (9, 2). Once we have these points, we can plot them and connect them to obtain the graph of \(f^{-1}(x)\).

Graph y=x

The graph of y=x is a straight line that passes through the origin (0,0) and has a positive gradient. The line y = x is the line of symmetry between any function and its inverse.