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Chapter 3.1

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.

3.3