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Q. 3.1.3

Graphing an Exponential Function

Graph: $g(x)=\left(\frac{1}{2}\right)^x$.

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.