Problem 104 - Introductory and Intermediate Algebra for College Students 4th

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

Function \(g(x)=-\log{x}\) is a vertical reflection of function \(f(x)=\log{x}\) over the x-axis.

Step 1: Graph the Function \(f(x) = \log{x}\)

To begin, graph the first function, \(f(x) = \log{x}\), in the viewing rectangle. This function represents a basic logarithm, so it should be fairly straightforward to plot. The function will start near negative infinity when \(x\) approaches 0 from the right and will pass through the point (1,0), growing slowly as \(x\) increases.

Step 2: Graph the Function \(g(x) = -\log{x}\)

In the same viewing rectangle, graph the second function \(g(x) = -\log{x}\). The negative sign means that this function will be a reflection of \(f(x)\) over the x-axis. It will start near positive infinity when \(x\) approaches 0 from the right, will pass through the point (1,0), and will decrease as \(x\) increases.

Step 3: Describe the Relationship

With both functions plotted, the relationship becomes clear. Function \(g(x)\) is a vertical reflection of function \(f(x)\) over the x-axis. This is clearly visible in the graph, as \(f(x)=\log{x}\) is flipped upside down to match with \(g(x)=-\log{x}\).