Chapter 7, Problem 7CA

a.

To determine

Whether the given function is increasing or decreasing for increasing values of $x$ .

Answer to Problem 7CA

The given function is increasing for increasing values of $x$ .

Explanation of Solution

Given Information: Given function is $f(x)=2{(3)}^x$ .

Calculation:

Putting different (increasing) values of $x$ in given function $f(x)=2{(3)}^x$ and arranging the results in table,

  $\begin{array}{llllll}x\hfill & -2\hfill & -1\hfill & 0\hfill & 1\hfill & 2\hfill \\ f(x)\hfill & \frac{2}{9}\hfill & \frac{2}{3}\hfill & 2\hfill & 6\hfill & 18\hfill \end{array}$

Clearly, $f(x)$ increases with increasing values of $x$ .

Also, it is clear from the graph that $f(x)$ increases with increasing $x$ .

Hence, the given function is increasing for increasing values of $x$ .

b.

To determine

To find: Any $x-\&y-$intercepts.

Answer to Problem 7CA

  $y-$ intercept is $x=2$ .

Explanation of Solution

Given Information: Given function is $f(x)=2{(3)}^x$ .

Calculation:

For $x-$ intercepts, putting $f(x)$

  $=0$ ,

  $\Rightarrow 2{(3)}^x=0$

  $\Rightarrow {(3)}^x=0$ , which is not true for any value of $x$ .

Also, from the graph, it is clear that the graph never intersects $x-axis.$

Hence, no $x-$ intercepts.

For $y-$ intercepts, putting $x=0$ in $f(x)=2{(3)}^x$ ,

  $f(x)=2{(3)}^0$

  $\Rightarrow f(x)=2(1)$

  $\Rightarrow f(x)=2$

Also, from the graph, it is clear that the graph intersects the $y-axis$ at $x=2$ .

Hence, $y-$ intercept is $x=2$ .