Chapter 5.3, Problem 51AYU

In problems $45-56,$ use the transformations to the graph each function. Determine the domain, range, horizontal asymptote and y-intercept of each function,

$f\left(x\right)=2+3^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

To determine

The graph of function $f(x)=2+3^{x/2}$ using transformation. Also, domain, range, horizontal asymptote, $y$- intercept of function.

Answer to Problem 51AYU

Solution:

The graph of $f(x)=2+3^{x/2}$ is

Domain of the function $f(x)=2+3^{x/2}$ is $\left(-\infty ,\infty \right)$

Range of function $f(x)=2+3^{x/2}$ is $\left(2,\infty \right)$

Horizontal asymptote of $f(x)=2+3^{x/2}$ is $y=2$

$y$- intercept of $f(x)=2+3^{x/2}$ is $3$

Explanation of Solution

Given information:

The function $f(x)=2+3^{x/2}$

Explanation:

To graph a function, begin with a graph of $f(x)=3^x$

Find some points satisfying $f(x)=3^x$

$\begin{array}{cccccc}x& -1& 0& 1& 2& -2\\ f\left(x\right)& \begin{array}{l}f\left(-1\right)=3^{-1}\\ =\frac{1}{3}\end{array}& \begin{array}{l}f\left(0\right)=3^0\\ =1\end{array}& \begin{array}{l}f\left(1\right)=3^1\\ =3\end{array}& \begin{array}{l}f\left(2\right)=3^2\\ =9\end{array}& \begin{array}{l}f\left(-2\right)=3^{-2}\\ =\frac{1}{9}\end{array}\\ \left(x,f\left(x\right)\right)& \left(-1,\frac{1}{3}\right)& \left(0,1\right)& \left(1,3\right)& \left(2,9\right)& \left(-2,\frac{1}{9}\right)\end{array}$

Plot these points on the graph.

The graph is

To graph a function $f(x)=3^{x/2}$, stretch the graph of $f(x)=3^x$ horizontally by $1/2$ unit.

The graph of function $f(x)=3^{x/2}$ is

To graph the function $f(x)=2+3^{x/2}$, shift the graph of $f(x)=3^{x/2}$ by $2$ units upwards.

The graph of $f(x)=2+3^{x/2}$ is

From the graph, domain of the function $f(x)=2+3^{x/2}$ is $\left(-\infty ,\infty \right)$

Range of the function is the set of values of the dependent variable for which a function is defined.

Therefore, range of $f(x)=3^{x-1}$ is $\left(2,\infty \right)$

As $x\to -\infty$, $f(x)$ approaches to $2$

Therefore, $f(x)=2+3^{x/2}$ has horizontal asymptote $y=2$

To find $y$- intercept, substitute $x=0$ in $f(x)=2+3^{x/2}$

$\Rightarrow f(0)=2+3^{0/2}=3$

Therefore, $y$- intercept is $3$