/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 Graph \(f(x)=\left(\frac{1}{4}\r... [FREE SOLUTION] | 91影视

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Chapter 3: 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
Both functions \(f(x) = (\frac{1}{4})^{x}\) and \(g(x) = log_{\frac{1}{4}} x\) are mirror images of each other with respect to the line \(y = x\). The two graphs show this fundamental property of a function and its inverse.

Step by step solution

01

Graph \(f(x)\)

Start graphing \(f(x) = (\frac{1}{4})^{x}\). This is an exponential function graph, with its y-intercept at 1 and decreasing since its base is between 0 and 1.
02

Graph \(g(x)\)

Then, graph \(g(x) = log_{\frac{1}{4}} x\). This is the inverse function of \(f(x)\), so you flip the graph of \(f(x)\) with respect to the line \(y = x\). The x-intercept is at 1 since for any \(b\), one property of logarithm functions is that \(log_{b}b = 1\). For \(x < 1\), the graph will also be decreasing.
03

Compare the Two Graphs

Finally, take a look at the two function graphs in the same Cartesian coordinate system. They are mirror images with respect to the line \(y = x\), because \(f(x)\) and \(g(x)\) are inverse functions of each other. This is a fundamental property when a function and its inverse are graphed on the same coordinate system.

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Most popular questions from this chapter

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x=10,100,1000\) \(10,000,100,000,\) and \(1,000,000 .\) Describe what happens to the expression as \(x\) increases.

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