Chapter 4, Problem 1MCCP

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

f(x) = 2^x and g(x) = 2^x – 3

To determine

The graph of the function $f\left(x\right)$ and $g\left(x\right)$ in the same rectangular coordinate system.

Answer to Problem 1MCCP

Solution: The graph of the function $f$ and $g$ is:

Explanation of Solution

Given: The function $f\left(x\right)=2^x$ and the second function is $g\left(x\right)=2^x-3$.

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

Find the some point of the provided function.

$x$	$f\left(x\right)=2^x$	$g\left(x\right)=2^x-3$
$-2$	$f\left(-2\right)=2^{-2}=\frac{1}{4}$	$g\left(-2\right)=2^{-2}-3=-\frac{11}{4}$
$-1$	$f\left(-1\right)=2^{-1}=\frac{1}{2}$	$g\left(-1\right)=2^{-1}-3=-\frac{5}{2}$
0	$f\left(0\right)=2^0=1$	$g\left(0\right)=2^0-3=-2$
1	$f\left(1\right)=2^1=2$	$g\left(1\right)=2^1-3=-1$

The horizontal asymptotes of the function $f$ is $y=0$. and the graph $g$ approaches the line $y=-3$ without touching its horizontal asymptotes.

The graph of the both functions is shown below as:

Both functions are rational functions have no horizontal asymptote.

The domain of the function $f$ and $g$ is the set of all real numbers or $\left(-\infty ,\infty \right)$ and the range is $\left(-3,\infty \right)$ or the set of all positive numbers.

Conclusion: The domain of the function $f$ and $g$ is the set of all real numbers or $\left(-\infty ,\infty \right)$ and the range of the function $g\left(x\right)$ is $\left(-3,\infty \right)$ and it is the set of all real numbers, which is greater than $-3$, and $f\left(x\right)$ is $\left(0,\infty \right)$ and it is the set of all positive numbers.