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Chapter 3: Problem 45

Graph \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\log _{\frac{1}{2}} x\) in the same rectangular coordinate system.

Short Answer

Expert verified
The graph of \(f(x)\) is a decreasing curve that passes through points like \((-2,4), (-1,2), (0,1), (1, \frac{1}{2}), (2, \frac{1}{4})\), and the graph of \(g(x)\) is an increasing curve passing through points like \((\frac{1}{4}, 2), (\frac{1}{2}, 1), (1, 0), (2, -1), (4, -2)\). The point of intersection of the two curves is at \(1,1\).

Step by step solution

01

Basic Definitions

Firstly, recognize that \(f(x)\) is an exponential function and \(g(x)\) is a logarithmic function. The base of the exponential function is \(\frac{1}{2}\) (which is less than 1), so the graph will be a decreasing curve. On the other hand, \(g(x)\) is a logarithm with base \(\frac{1}{2}\), so its graph would be a mirror image of the graph of \(f(x)\) with respect to the line \(y = x\)
02

Plotting Graph of \(f(x)\)

A plot of \(f(x)\) can start from \(-2\) and end at \(2\). For instance, when \(x = -2, -1, 0, 1, 2\), \(f(x)\) gives \(4, 2, 1, \frac{1}{2}, \frac{1}{4}\), respectively. Connect these points to get a decreasing curve on the graph.
03

Plotting Graph of \(g(x)\)

For \(g(x)\), you have to remember as the logarithm is only defined for \(x > 0\), exclude \(0\) from the x-values. When \(x = \frac{1}{4}, \frac{1}{2}, 1, 2, 4\), \(g(x)\) gives \(2, 1, 0, -1, -2\), respectively. Connect these points to get an increasing curve on the graph.
04

Finalize the Coordinate System

You should mark all important points and label the axes. Remember that the two graphs intersect at the point \((1,1)\) because \(f(g(x)) = x\) and \(g(f(x)) = x\)

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