Chapter 4, Problem 1MC

In Exercises 1-5, graph f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. Give each function’s domain and range.

1. f(x) = 2^x and g(x) = 2^x − 3

To determine

To sketch: The graph of the functions $f\left(x\right)=2^x$ and $g\left(x\right)=2^x-3$, in the same rectangular coordinate system and find the asymptotes, domain and range of the functions

Answer to Problem 1MC

The asymptote of both the curves $f\left(x\right)=2^x$ and $g\left(x\right)=2^x-3$ is $y=0$ and $y=-3$.

The domain of the curves is $\left(-\infty ,\infty \right)$ and ranges of the curves are $\left(0,\infty \right)$ and $\left(-3,\infty \right)$ respectively.

Explanation of Solution

Consider the given functions $f\left(x\right)=2^x$ and $g\left(x\right)=2^x-3$.

Use online graphing calculator to draw the graphs of the functions $f\left(x\right)=2^x$ and $g\left(x\right)=2^x-3$ as shown below in Figure 1.

From Figure 1, it is observed that the curve $f\left(x\right)=2^x$ goes up to the right and is an increasing function.

It approaches negative x-axis as $x\to -\infty$ and hence x-axis is a horizontal asymptote.

The curve $g\left(x\right)=3^{-x}$ goes up to the right and is an increasing function.

It approaches $y=-3$ as $x\to -\infty$ and hence $y=-3$ is a horizontal asymptote.

Note that the domain of the curves is $\left(-\infty ,\infty \right)$.

And the range of the curves is $\left(0,\infty \right)$ and $\left(-3,\infty \right)$ respectively.

Thus, the asymptote of both the curves $f\left(x\right)=2^x$ and $g\left(x\right)=2^x-3$ is $y=0$ and $y=-3$.

The domain of the curves is $\left(-\infty ,\infty \right)$ and ranges of the curves are $\left(0,\infty \right)$ and $\left(-3,\infty \right)$ respectively.