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Chapter 4: Problem 43

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

Short Answer

Expert verified
The graph of the exponential function \(f(x)=4^{x}\) starts at the point (0,1) and increases quickly as x increases. The graph of the logarithmic function \(g(x)=\log _{4} x\) passes through the point (1,0) and increases slowly as x increases. The two graphs are mirror images of each other across the line y=x.

Step by step solution

01

Understanding the given functions

The given functions are \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\). \(f(x)=4^{x}\) is an exponential function. It will always be positive and will pass through the point (0,1). As x increases, \(f(x)\) increases quickly. \(g(x)=\log _{4} x\) is the inverse function of the the exponential function \(f(x)=4^{x}\).
02

Sketching the graph of \(f(x)=4^{x}\)

To begin the sketch of this function, plot the point (0,1) because any non-zero number raised to the power of 0 is 1. Then choose a few simple x-values to determine the shape of the curve. For example, when x = 1, \(f(x) = 4^{1} = 4\), and when x = -1, \(f(x) = 4^{-1} = 0.25\). The curve should pass through the points (-1, 0.25), (0,1) and (1,4), to start with. As x increases, \(4^{x}\) gets larger. Thus, the curve goes up steeply. As x decreases, \(4^{x}\) becomes a small positive number. Therefore, the curve closely approaches but never reaches the x-axis.
03

Sketching the graph of \(g(x)=\log _{4} x\)

Since the logarithmic function is the inverse function of the exponential function, the graph of \(g(x)=\log _{4} x\) is obtained by reflecting the graph of \(f(x)=4^{x}\) about the line y=x. More specifically, for every point (x, y) on the graph of \(f(x)=4^{x}\), there is a corresponding point (y, x) on the graph of \(g(x)=\log _{4} x\). Since \(f(x)=4^{x}\) passes through (0,1), \(g(x)=\log _{4} x\) will pass through (1,0). When x = 4 in \(g(x)=\log _{4} x\), \(g(x) = \log _{4} 4 = 1\). So (4,1) is also on the curve. As x increases, \(\log _{4} x\) slowly increases. As x moves toward zero, \(\log _{4} x\) drops sharply. So the graph closely approaches but never crosses the y-axis.
04

Combining the graphs

After obtaining the individual graphs of \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\), put them together in the same rectangular coordinate system. Since these two functions are inverses, their graphs are mirror images of each other across the line y=x.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)

Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, \(f(t),\) after \(t\) months was modeled by the function $$f(t)=88-15 \ln (t+1), \quad 0 \leq t \leq 12$$ a. What was the average score on the original exam? b. What was the average score after 2 months? 4 months? 6 months? 8 months? 10 months? one year? c. Sketch the graph of \(f\) (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students.

Find the domain of each logarithmic function. $$f(x)=\log \left(\frac{x-2}{x+5}\right)$$

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Each group member should consult an almanac, newspaper. magazine, or the Internet to find data that can be modeled by exponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?
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