Problem 39 - College Algebra

Question

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

Step-by-Step Solution

Answer

The important relationship to note from the graphing of these two functions comes from the properties of logarithmic and exponential functions. As seen in the graphical representation, the two graphs are mirror images of each other along the line \(y=x\). This is due to the fact that logarithmic and exponential functions are inverses of each other.

Step 1: Graph the exponential function

Start by graphing the function \(f(x)=4^{x}\). Remember that any exponential function will always pass through the point (0,1) because anything raised to the power of 0 is 1. Also, because the base 4 is greater than 1, the function will be increasing. Thus, to the right, the graph rises, and to the left, it gets closer and closer to the x-axis but never touches or crosses it.

Step 2: Graph the logarithmic function

Now, graph the logarithmic function \(g(x)=\log _{4} x\). The point (4,1) will always be on the logarithmic curve because \(\log_4 4 = 1\). Also, since logarithms are undefined for zero and negative numbers, the graph will only exist to the right of the y-axis. The graph of \(g(x)=\log _{4} x\) will resemble somewhat of a mirror image of the exponential function along the line \(y=x\), thanks to the logarithmic and exponential functions being inverses of each other.

Step 3: Overlay both graphs onto a single coordinate system

Upon completing the graphs of both functions, overlay both onto the same rectangular coordinate system. This will allow for a direct comparison between the functions and a clear visual representation of their properties and relationships, notably that they are mirror images of each other across the line \(y=x\) in the coordinate.