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Chapter 4: Problem 46

Graph \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\log _{\frac{1}{4}} x\) in the same rectangular coordinate system.

Short Answer

Expert verified
The functions \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\log _{\frac{1}{4}} x\) are inverses of each other. The graph of \(f(x)\) is decreasing and asymptotic to the \(x\)-axis. The graph of \(g(x)\) is decreasing and asymptotic to the \(y\)-axis. They intersect at \((1,0)\).

Step by step solution

01

Graph of \(f(x)=\left(\frac{1}{4}\right)^{x}\)

To graph the exponential function, first identify a few key points. For \(x=0\), \(f(x)=1\). As \(x\) increases, \(f(x)\) decreases and approaches 0, and as \(x\) decreases, \(f(x)\) increases without bound. The graph passes through \((0,1)\) and is asymptotic to the \(x\)-axis.
02

Graph of \(g(x)=\log _{\frac{1}{4}} x\)

To graph the logarithmic function, note that \(g(x)=\log _{\frac{1}{4}} x\) is the inverse of \(f(x)\). Therefore, its graph is a reflection of the graph of \(f(x)\) in the line \(y=x\). The graph passes through \((1,0)\), approaches the \(y\)-axis as an asymptote as \(x\) decreases, and decreases without bound as \(x\) increases.
03

Combine the two graphs

Draw the graphs of \(f(x)\) and \(g(x)\) on the same set of axes. The \(x\)-axis is a horizontal asymptote for \(f(x)\) and the \(y\)-axis is a vertical asymptote for \(g(x)\). The two graphs intersect at \((1,0)\).

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Most popular questions from this chapter

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the graph of \(g(x)=\log _{2} x\).

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

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