Problem 83 - Precalculus with Limits

Question

Determine whether the statement is true or false. Justify your answer. The graph of \(f(x)=\log _{6} x\) is a reflection of the graph of \(g(x)=6^{x}\) in the \(x\) -axis.

Step-by-Step Solution

Answer

No, the statement is false. The graph of \(f(x) = \log_{6}x\) is not the reflection of the graph of \(g(x) = 6^{x}\) in the x-axis. Instead, the two graphs are inverses of each other.

Step 1: Understanding the functions

The general shape of an exponential function is increasing if the base is greater than 1, and decreasing if the base is between 0 and 1. The general shape of a logarithmic function is increasing for all positive values of \(x\). The graph of \(f(x) = \log_{6}x\) is the inverse of the graph of \(g(x) = 6^x\), but not its reflection.

Step 2: Analysing the proposition

If \(\log_{6}x\) would be the reflection of \(6^{x}\), that would mean for every \(x\) such that \(f(x) = a\), we would have \(g(x) = -a\). However, this is not true. The graph of \(f(x) = \log_{6}x\) is the inverse of \(6^{x}\), meaning that for a value \(a\), \(f(x) = a\) would mean \(g(a) = x\), not \(g(x) = -a\). Therefore, the statement is not correct: The graph of \(f(x) = \log_{6}x\) is not the reflection of the graph of \(g(x) = 6^{x}\) across the x-axis.