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Chapter 3: Problem 49

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. $$f(x)=3^{x} \text { and } g(x)=\frac{1}{3} \cdot 3^{x}$$

Short Answer

Expert verified
Both functions \(f(x)\) and \(g(x)\) are graphed with their respective increase as \(x\) increases. They both share a horizontal asymptote at \(y=0\).

Step by step solution

01

Plot function \(f(x)\)

Plot the graph of \(f(x)=3^{x}\). This is an exponential function with a positive base greater than 1, so it increases as \(x\) increases. It will also have a horizontal asymptote at \(y=0\).
02

Plot function \(g(x)\)

Next, plot the graph of \(g(x)=\frac{1}{3}\cdot3^{x}\). This is a transformation of the function \(f(x)\) where we have multiplied it by \(1/3\). So, it will also be an exponential function that increases as \(x\) increases but it will be less steep than \(f(x)\). It has also a horizontal asymptote at \(y=0\).
03

Asymptotes of \(f(x)\) and \(g(x)\)

For both functions, there is a horizontal asymptote at \(y=0\) (the x-axis). This is the line that the functions approach but never quite reach.
04

Confirming with a graphing utility

To confirm the accuracy of our hand-drawn graphs, plot both functions on a graphing utility and compare. The outputs should match the predictions made in Steps 1 and 2.

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