Chapter 6, Problem 13CT

To determine

To compare the graph of $f(x)=2{\left(6\right)}^x$ to graph of $g(x)={\left(6\right)}^x$ and describe the domain and range of $f$ .

Answer to Problem 13CT

The functions $f(x)=2{\left(6\right)}^x$ is vertically stretched of $f(x)$ by a factor of $2$ units with domain $=\left(-\infty ,\infty \right)$ and range $=\left[0,\infty \right)$

Explanation of Solution

Given information :

Given functions $f(x)=2{\left(6\right)}^x$ and $g(x)={\left(6\right)}^x$

As we know that the function $f(x)=kg(x)$ is vertically stretched of $f(x)$ by a factor of $k$ units, if $k>1$

By comparing $f(x)$ and $g(x)$ , we get $k=2>1$ .

Thus, the functions $f(x)=2{\left(6\right)}^x$ is vertically stretched of $f(x)$ by a factor of $2$ units.

Domain is the set of all possible values of $x$ such that $f(x)$ is well defined.

Since $f(x)=2{\left(6\right)}^x$ is well defined for every real value of $x$ .

So domain of $f(x)=2{\left(6\right)}^x$ is $\left(-\infty ,\infty \right)$

Similar way, the range is set of all possible values of $y$ such $y=f(x)$ .

since $f(x)\ge 0\forall x\in R(\text{Real}\text{number}\text{set})$

So, range of $f(x)$ is $\left[0,\infty \right)$

Hence, the functions $f(x)=2{\left(6\right)}^x$ is vertically stretched of $f(x)$ by a factor of $2$ units with domain $=\left(-\infty ,\infty \right)$ and range $=\left[0,\infty \right)$ .