Problem 130 - College Algebra Essentials

Question

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\log x, g(x)=-\log x$$

Step-by-Step Solution

Answer

The graph of \(g(x)=-\log x\) is the reflection of the graph of \(f(x)=\log x\) over the x-axis.

Step 1: Sketch the graph of \(f(x)=\log x\)

Begin by sketching the graph of \(f(x)=\log x\). The graph of \(f(x)=\log x\) is a smooth curve that crosses the x-axis at x=1 and then increases without bound to the right while it moves closer to the x-axis but never touches or crosses it, as \(x\) moves to the left.

Step 2: Sketch the graph of \(g(x)=-\log x\)

The graph of \(g(x)=-\log x\) can be obtained from the graph of \(f(x)=\log x\) by reflecting the latter in the x-axis. This is because the negative sign only affects the y-coordinates of the points on \(f\), leaving the x-coordinates unchanged.

Step 3: Compare the graphs

After sketching the graphs of both functions, compare them. You will notice that the graph of \(g(x)=-\log x\) is simply the reflection of the graph of \(f(x)=\log x\) over the x-axis.