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Chapter 3: Problem 17

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. $$f(x)=\left(\frac{1}{2}\right)^{x}$$

Short Answer

Expert verified
The function \(f(x)=(\frac{1}{2})^{x}\) gives a decreasing graph, starting from positive y-values for negative x-values, and getting closer to the x-axis as x increases. This is shown by the calculated table and the plotted graph.

Step by step solution

01

Construct the Table of Values

To construct a table of values, choose a range of x-values and compute the corresponding y-values according to the function \(f(x)=(\frac{1}{2})^{x}\). For example, if we choose x-values between -3 and 3, we get the following pairs:x=-3, y= \(f(-3)=(\frac{1}{2})^{-3}=8\)x=-2, y= \(f(-2)=(\frac{1}{2})^{-2}=4\)x=-1, y= \(f(-1)=(\frac{1}{2})^{-1}=2\)x=0, y= \(f(0)=(\frac{1}{2})^{0}=1\)x=1, y= \(f(1)=(\frac{1}{2})^{1}=\frac{1}{2}\)x=2, y= \(f(2)=(\frac{1}{2})^{2}=\frac{1}{4}\)x=3, y= \(f(3)=(\frac{1}{2})^{3}=\frac{1}{8}\)
02

Plot the Pair of Values on Graph

Using the above pairs of x-y values, the points can be plotted on a graph. Note how as the x value increases, the y value decreases - indicating the function is decreasing.
03

Sketch the Graph

The graph of the function \(f(x)=(\frac{1}{2})^{x}\) starts at high positive y-values for negative x-values, and asymptotically approaches the x-axis towards the right, never reaching it as the increase of the x-values gives smaller y-values.

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