## 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.