Chapter 4, Problem 1T

(a)

To determine

To sketch: The graph of $f\left(x\right)=2^{-x}+4$, and states its domain, range, and asymptotes and also represents the $x$ and $y$- intercepts on the graph.

Answer to Problem 1T

The domain of the function $f\left(x\right)=2^{-x}+4$ is $\left(-\infty ,\infty \right)$ and range $\left[4,\infty \right)$ and asymptotes parallel to $x$-axis is $y=4$ and the intercept on the $y$-axis is $y=5$.

Explanation of Solution

Graph:

Use online graphing calculator and draw the graph of $f\left(x\right)=2^{-x}+4$ as shown below in Figure 1.

From Figure 1, it is noticed that the graph of $f\left(x\right)=2^{-x}+4$ intersect the $y$-axis at $y=5$, and defined every value of $x$ on the real line.

The domain of the function is the set of all possible values of $x$ for which the function $f$ defines.

Thus, the domain of the function $f\left(x\right)=2^{-x}+4$ is $\left(-\infty ,\infty \right)$.

The range of the function is the set of all pre-images of $f$ at $x$.

The value of the function $f\left(x\right)$ is equal to 4 as $x\to \infty$.

Thus, the range of the function $f\left(x\right)=2^{-x}+4$ is $\left[4,\infty \right)$.

As $x$ approaches to $\infty$, the line $y=4$ is the tangent to the curve at $x\to \infty$.

Thus, the asymptotes parallel to the $x$-axis is $y=4$.

The graph of the function cut first time on the $y$-axis at a distance $y=5$, this means that the graph of $f\left(x\right)$ has the intercept on the $y$-axis of $y=5$.

Therefore, the domain of the function $f\left(x\right)=2^{-x}+4$ is $\left(-\infty ,\infty \right)$ and range $\left[4,\infty \right)$ and asymptotes parallel to $x$-axis is $y=4$ and the intercept on the $y$-axis is $y=5$.

(b)

To determine

To sketch: The graph of $g\left(x\right)={\log }_3\left(x+3\right)$, and states its domain, range, and asymptotes and also represents the $x$- and $y$- intercepts on the graph.

Answer to Problem 1T

The domain of the function $g\left(x\right)={\log }_3\left(x+3\right)$ is $\left(-3,\infty \right)$ and range $\left[-3.657,\infty \right)$ and the asymptotes parallel to $y$-axis is $x=-3$ and the intercept on the axes are $x=-2$ and $y=1$.

Explanation of Solution

Graph:

Use online graphing calculator and draw the graph of $g\left(x\right)={\log }_3\left(x+3\right)$ as shown below in Figure 2.

From Figure 2, it is noticed that the graph of $g\left(x\right)={\log }_3\left(x+3\right)$ cut the $y$-axis at $y=1$, and $x$-axis at $x=-2$ defined every for $x>-3$.

The domain of the function is the set of all possible values of $x$ for which the function $g$ defines.

Thus, the domain of the function $g\left(x\right)={\log }_3\left(x+3\right)$ is $\left(-3,\infty \right)$.

Substitute, $x=-2.982$ in $g\left(x\right)$,

$\begin{array}{c}g\left(-2.982\right)={\log }_3\left(-2.982+3\right)\\ ={\log }_3\left(0.018\right)\\ =-3.657\end{array}$

The range of the function is the set of all pre-images of $g$ at $x$.

And the value of the function $g\left(x\right)$ approaches to $\infty$ as $x\to \infty$.

Thus, the range of the function is $\left[-3.657,\infty \right)$.

The function $g\left(x\right)$ meet to the minus infinity at $x=-3$.

Thus, the function $g\left(x\right)$ has asymptotes parallel to $y$-axis is $x=-3$.

The graph of the function cut first time on the $y$-axis at a distance $y=1$, and on the $x$-axis at a distance $x=-2$.

The intercept on the $x$-axis is $x=-2$ and on the $y$-axis is $y=1$.

Therefore, the domain of the function $g\left(x\right)={\log }_3\left(x+3\right)$ is $\left(-3,\infty \right)$ and range $\left[-3.657,\infty \right)$ and the asymptotes parallel to $y$-axis is $x=-3$ and the intercept on the axes are $x=-2$ and $y=1$.