Problem 43 - College Algebra Essentials

Question

Graph \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graph of the exponential function \(f(x)=4^{x}\) starts at the point (0,1) and increases quickly as x increases. The graph of the logarithmic function \(g(x)=\log _{4} x\) passes through the point (1,0) and increases slowly as x increases. The two graphs are mirror images of each other across the line y=x.

Step 1: Understanding the given functions

The given functions are \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\). \(f(x)=4^{x}\) is an exponential function. It will always be positive and will pass through the point (0,1). As x increases, \(f(x)\) increases quickly. \(g(x)=\log _{4} x\) is the inverse function of the the exponential function \(f(x)=4^{x}\).

Step 2: Sketching the graph of \(f(x)=4^{x}\)

To begin the sketch of this function, plot the point (0,1) because any non-zero number raised to the power of 0 is 1. Then choose a few simple x-values to determine the shape of the curve. For example, when x = 1, \(f(x) = 4^{1} = 4\), and when x = -1, \(f(x) = 4^{-1} = 0.25\). The curve should pass through the points (-1, 0.25), (0,1) and (1,4), to start with. As x increases, \(4^{x}\) gets larger. Thus, the curve goes up steeply. As x decreases, \(4^{x}\) becomes a small positive number. Therefore, the curve closely approaches but never reaches the x-axis.

Step 3: Sketching the graph of \(g(x)=\log _{4} x\)

Since the logarithmic function is the inverse function of the exponential function, the graph of \(g(x)=\log _{4} x\) is obtained by reflecting the graph of \(f(x)=4^{x}\) about the line y=x. More specifically, for every point (x, y) on the graph of \(f(x)=4^{x}\), there is a corresponding point (y, x) on the graph of \(g(x)=\log _{4} x\). Since \(f(x)=4^{x}\) passes through (0,1), \(g(x)=\log _{4} x\) will pass through (1,0). When x = 4 in \(g(x)=\log _{4} x\), \(g(x) = \log _{4} 4 = 1\). So (4,1) is also on the curve. As x increases, \(\log _{4} x\) slowly increases. As x moves toward zero, \(\log _{4} x\) drops sharply. So the graph closely approaches but never crosses the y-axis.

Step 4: Combining the graphs

After obtaining the individual graphs of \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\), put them together in the same rectangular coordinate system. Since these two functions are inverses, their graphs are mirror images of each other across the line y=x.