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Chapter 3: Problem 48

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. $$f(x)=3^{x} \text { and } g(x)=-3^{x}$$

Short Answer

Expert verified
The graphs of \(f(x)=3^{x}\) and \(g(x)=-3^{x}\) show that they have the same horizontal asymptote at \(y=0\). The function \(f(x)=3^{x}\) grows rapidly for positive \(x\), while \(g(x)=-3^{x}\) decreases rapidly for positive \(x\). The use of a graphing utility confirms these observations.

Step by step solution

01

Understand the Exponential Function

The function \(f(x)=3^{x}\) is an exponential function with base 3. For positive \(x\), it will grow rapidly, and for negative \(x\), it will approach 0. Similarly, \(g(x)=-3^{x}\) is the reflection of \(f(x)\) along the x-axis. These functions do not have vertical asymptotes because the domain is all real numbers.
02

Identify the Horizontal Asymptotes

Horizontal asymptotes occur when the function approaches a particular value as \(x\) approaches either positive or negative infinity. For \(f(x)=3^{x}\), as \(x\) approaches negative infinity, \(f(x)\) approaches 0 from above. Therefore, the horizontal asymptote is \(y=0\). Similarly, for \(g(x)=-3^{x}\), as \(x\) approaches negative infinity, \(g(x)\) approaches 0 from below. Therefore, the horizontal asymptote is \(y=0\).
03

Graph the Functions

Draw and label the x and y axes. Then draw the graphs \(f(x)=3^{x}\) and \(g(x)=-3^{x}\) such that they go through the point (0,1) and (0,-1) respectively. They should approach, but never cross, the line \(y=0\) showing that \(y=0\) is a horizontal asymptote for both functions.
04

Confirm the Graphs

Use a graphing utility to confirm the hand-drawn graphs. The graphs should cover the three important aspects: where they cross the y-axis, their general behavior as \(x\) gets larger or smaller, and their asymptotes.

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