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Chapter 3: Problem 35

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. $$f(x)=6^{x}, g(x)=\log _{6} x$$

Short Answer

Expert verified
The graphs of \(f(x)=6^{x}\) and \(g(x)=\log_{6}x\) are reflections of each other about the line \(y=x\). The exponential function rises to the right for positive x values and approaches zero for negative x values, while the logarithmic function rises to the right for positive x values and will reach negative infinity as x approaches zero.

Step by step solution

01

Understand the Functions

The function \(f(x)=6^{x}\) is an exponential function and it will have a graph that rises to the right for positive x values and approaches zero for negative x values. The function \(g(x)=\log_{6}x\) is a logarithmic function, and the graph will rise to the right for positive x values and will reach negative infinity as x approaches zero.
02

Draw the Graph of the Exponential Function

First, sketch the graph of \(f(x)=6^{x}\). To do this, take a few key points such as \(x=-2, -1, 0, 1, 2\) and calculate their corresponding \(f(x)\) values to plot them. Then, connect these points smoothly to sketch the graph. It should show a graph that increases rapidly for positive x and slowly approaches 0 for negative x.
03

Draw the Graph of the Logarithmic Function

Next, sketch the graph of \(g(x)=\log_{6}x\). Just like in the previous step, take the same key points. However, for the logarithmic function, keep in mind that it only takes positive values of x. Plot those values to get the points on the graph for \(g(x)\). Then, connect these points smoothly to sketch the graph. It should show a graph that increases slowly for larger x values and reaches negative infinity as x approaches zero.
04

Comparing the Graphs

Looking at the graphs of both the functions, we see that the graphs are reflections of each other about the line \(y=x\). This is a general property of logarithmic and exponential functions that are inverses of each other.

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Most popular questions from this chapter

Students in a mathematics class took an exam and ther took a retest monthly with an equivalent exam. The average scores for the class are given by the human memory model \(f(t)=80-17 \log (t+1), \quad 0 \leq t \leq 12\) where \(t\) is the time in months.(a) Use a graphing utility to graph the model over the specified domain. (b) What was the average score on the original \(\operatorname{exam}(t=0) ?\)

Function \(\quad\) Value $$ f(x)=\ln x \quad x=18.42$$

Limit of a Function (a) Complete the table for the function \(f(x)=(\ln x) / x\)\begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 5 & 10 & 10^{2} & 10^{4} & 10^{6} \\ \hline f(x) & & & & & & \\ \hline \end{array}

Write the logarithmic equation in exponential form. $$\ln 7=1.945 \ldots$$

Graphical Analysis Use a graphing utility to graph \(f\) and \(g\) in the same viewing window and determine which is increasing at the greater rate as \(x\) approaches + \(\infty\). What can you conclude about the rate of growth of the natural logarithmic function? (a) \(f(x)=\ln x, \quad g(x)=\sqrt{x}\) (b) \(f(x)=\ln x, \quad g(x)=\sqrt[4]{x}\)
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