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 function’s domain and range.

$f\left(x\right)=2^x\text{ and }g\left(x\right)=2^x-3$

To determine

To calculate: Domain, range and equation of asymptotes for given functions $f\left(x\right)=2^x,g(x)=2^x-3$.

Answer to Problem 1MCCP

Solution:

Asymptote of $\underset{\_}{f:y=3}$.

Asymptote of $\underset{\_}{g:y=-3}$.

Domain of $f$ $=$ Domain of $g=\left(-\infty ,\infty \right)$.

Range of $\underset{\_}{f=\left(0,\infty \right)}$, Range of $\underset{\_}{g=\left(-3,\infty \right)}$.

Explanation of Solution

Given: The given functions: $f\left(x\right)=2^x,g(x)=2^x-3$

Calculation:

Create a table of co-ordinates for the given function $f\left(x\right)=2^x$

x	-2	-1	0	1	2
$f(x)=2^x$	$\begin{array}{l}f(-2)=2^{-2}\\ =\frac{1}{2^2}=\frac{1}{4}\end{array}$	$\begin{array}{l}f(-1)=2^{-1}\\ =\frac{1}{2^1}=\frac{1}{2}\end{array}$	$\begin{array}{l}f(0)=2^0\\ =1\end{array}$	$\begin{array}{l}f(1)=2^1\\ =2\end{array}$	$\begin{array}{l}f(2)=2^2\\ =4\end{array}$

Plot these points, connecting them with a curve for graph of $f\left(x\right)=2^x$.

To plot the graph of a given function $g(x)=b^x-c=2^x-3$, shift the graph of $f(x)=b^x=2^x$ downwards $c=3$ units.

To determine

To graph: $f\left(x\right)=2^x,g(x)=2^x-3$

Explanation of Solution

Given: The given functions: $f\left(x\right)=2^x,g(x)=2^x-3$

Graph:

Interpretation:

The graph for $f\left(x\right)=2^x$ never touches negative portion of x-axis. Hence x-axis or $y=0$ is asymptote.

The asymptote for $g\left(x\right)=2^x-3$ is $y=-3$.

Domain of $f(x)=b^x\text{and}g(x)=b^x-c$ consists of all real numbers $\left(-\infty ,\infty \right)$.

Hence domain of $f\left(x\right)=2^x=$ domain of $g\left(x\right)=2^x-3$ $=\left(-\infty ,\infty \right)$.

Range of $f(x)=b^x$ consists of all positive real numbers $\left(0,\infty \right)$.

Hence range of $f\left(x\right)=2^x$ is $\left(0,\infty \right)$.

From the graph, range of $g\left(x\right)=2^x-3$ is $\left(-3,\infty \right)$.