Chapter 4.1, Problem 2E

To determine

To match: The given exponential functions with the given graphs.

Answer to Problem 2E

The correct matches are $\left(\text{a}\right)\to III$ , $\left(\text{b}\right)\to I$ , $\left(\text{c}\right)\to II$ and $\left(\text{d}\right)\to IV$ .

Explanation of Solution

Given:

The given exponential functions are $\left(\text{a}\right)f\left(x\right)=2^x$ , $\left(\text{b}\right)f\left(x\right)=2^{-x}$ , $\left(\text{c}\right)f\left(x\right)=-2^x$ and $\left(\text{d}\right)f\left(x\right)=-2^{-x}$ .

The given graphs are:

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Calculation:

Consider the function $f\left(x\right)=2^x$ . Plot the function in a graphing utility. The obtained graph is shown in the image below:

  

Figure 1

Graph in Figure 1 matches the graph marked III. So, (a) matches with III.

Now consider the function $f\left(x\right)=2^{-x}$ . The graph of $f\left(x\right)=2^{-x}$ is the graph of $2^x$ reflected in the I-axis that resembles that graph labeled I. So, (b) matches with I.

Consider the function $f\left(x\right)=-2^x$ . The graph of $f\left(x\right)=-2^x$ is the graph of $2^x$ reflected in the x-axis that resembles that graph labeled II. So, (c) matches with II.

Finally, consider the function $f\left(x\right)=-2^{-x}$ . The graph of $f\left(x\right)=-2^{-x}$ is the graph of $2^x$ first reflected in the y-axis and then through the x-axis that resembles that graph labeled IV. So, (d) matches with IV.

So, the correct matches are $\left(\text{a}\right)\to III$ , $\left(\text{b}\right)\to I$ , $\left(\text{c}\right)\to II$ and $\left(\text{d}\right)\to IV$ .