Problem 47 - College Algebra Essentials

Question

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$f(x)=3^{x} \text { and } g(x)=3^{-x}$$

Step-by-Step Solution

Answer

The graphs of \(f(x)=3^{x}\) and \(g(x)=3^{-x}\) both have the horizontal asymptote at \(y=0\). There are no vertical asymptotes for either functions. The behavior of the graphs are confirmed by a graphing utility.

Step 1: Graph functions

On a rectangular coordinate system, plot two functions: \(f(x)=3^{x}\) and \(g(x)=3^{-x}\). Observe the behavior of both functions as \(x\) increases or decreases. \(f(x)\) increases as \(x\) increases, \(g(x)\) decreases.

Step 2: Identify asymptotes

For \(f(x)=3^{x}\), as \(x\) tends to negative infinity, \(f(x)\) approaches 0. So, the horizontal asymptote is \(y=0\). Similarly for \(g(x)=3^{-x}\), as \(x\) tends to positive infinity, \(g(x)\) also approaches 0. So, its horizontal asymptote is also \(y=0\). Neither function has vertical asymptotes as they have defined values for all \(x\).

Step 3: Confirm graphs using a graphing utility

Use a graphing utility to plot these functions to confirm the hand-drawn plots and asymptote analysis. The graphs of \(f(x)=3^{x}\) and \(g(x)=3^{-x}\) should show the functions approaching but never reaching the horizontal line \(y=0\) as expected.