Chapter 3.1
Q. 3.1.3
Q. 3.1.3
Graphing an Exponential Function
Graph: g(x)=\left(\frac{1}{2}\right)^x.
Step-by-Step
Verified Solution
We begin by setting up a table of coordinates. We compute the function values by noting that
g(x)=\left(\frac{1}{2}\right)^x=\left(2^{-1}\right)^x=2^{-x}.
| x | g(x)=\left(\frac{1}{2}\right)^x \text { or } 2^{-x} |
| -3 | g(-3)=2^{-(-3)}=2^3=8 |
| -2 | g(-2)=2^{-(-2)}=2^2=4 |
| -1 | g(-1)=2^{-(-1)}=2^1=2 |
| 0 | g(0)=2^{-0}=1 |
| 1 | g(1)=2^{-1}=\frac{1}{2^1}=\frac{1}{2} |
| 2 | g(2)=2^{-2}=\frac{1}{2^2}=\frac{1}{4} |
| 3 | g(3)=2^{-3}=\frac{1}{2^3}=\frac{1}{8} |
We plot these points, connecting them with a continuous curve. Figure 3.3 shows the graph of g(x)=\left(\frac{1}{2}\right)^x. This time the graph approaches, but never touches, the positive portion of the x-axis. Once again, the x-axis, or y = 0, is a horizontal asymptote. The range consists of all positive real numbers.
