Problem 124 - College Algebra Essentials

Question

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the graph of \(g(x)=\log _{2} x\).

Step-by-Step Solution

Answer

The graph of \( g(x) = \log_2x \) is obtained by reflecting the graph of \( f(x) = 2^{x} \) over the line \( y = x \). Key points in the plot of \( f(x) = 2^{x} \) transpose their x and y coordinates in the plot of the inverse function. Lastly, the graph of \( g(x) = \log_2x \) is asymptotic to the y-axis (x = 0).

Step 1: Reflect the graph

Reflect the graph of \( f(x) = 2^{x} \) over the line \( y = x \). This reflected graph will give the general shape plotted of \( g(x) = \log_2x \). This reflection occurs because \( x = 2^{y} \) implies \( y = \log_2x \), so each point (x, y) on the \( f(x) = 2^{x} \) graph has a corresponding point (y, x) on the \( g(x) = \log_2x \) graph.

Step 2: Establish Key Points

It's important to identify and label several key points from the graph of \( f(x) = 2^{x} \) that will become key points on the graph of \( g(x) = \log_2x \). For instance, the point (1, 2) on \( f(x) = 2^{x} \) becomes the point (2, 1) on \( g(x) = \log_2x \). Similarly, the point (0, 1) on \( f(x) = 2^{x} \) becomes the point (1, 0) on \( g(x) = \log_2x \).

Step 3: Sketching the Graph

Now having determined the reflection and marked down the key coordinates, you can sketch the new transformation \( g(x) = \log_2x \) in a new contrasting color. Make sure the graph is asymptotic to \( x = 0 \) (the y-axis) because the logarithm function is undefined at x = 0.