Chapter 3.1, Problem 45E

Solution

To graph: The given function and to analyze it.

  

The domain is $\left(-\infty ,\infty \right)$ .

The range is $\left(0,\infty \right)$ .

It is a continuous function for all real numbers.

The function is always increasing.

No symmetry.

Bounded below, but not above.

No local extrema.

Horizontal asymptote is $y=0$ .

No vertical asymptote.

End behavior: $\underset{x\to -\infty }{\lim }f\left(x\right)=0,\underset{x\to \infty }{\lim }f\left(x\right)=\infty$ .

Given:

  $f\left(x\right)=3\cdot 2^x$

Calculation:

Comparing $f\left(x\right)=3\cdot 2^x$ with $f\left(x\right)=a\cdot b^x$ , it follows that

  $a=3,b=2$ .

Since $b>1$ , therefore it is an increasing function.

Now, the graph is

  

The given function is well defined for all real numbers, therefore the domain is $\left(-\infty ,\infty \right)$ .

From the graph, it can be seen that the $y$ -values are ranging from 0 to $\infty$ .

Therefore, the range is $\left(0,\infty \right)$ .

From the graph, it can be seen that the curve has no breaks or jumps; therefore it is a continuous function for all real numbers.

From the graph, it can be seen that the value of $y$ increases as the value of $x$ increases from $-\infty$ to $\infty$ .

So, the function is always increasing.

The curve is neither symmetric with the $x$ -axis nor with the $y$ -axis.

So the curve has no symmetry.

Bounded below, but not above.

There are no turning points for the given curve, so there is no local extrema.

Horizontal asymptote is $y=0$ .

No vertical asymptote.

End behavior:

From the graph, it can be seen that $y\to 0$ as $x\to -\infty$ and $y\to \infty$ as $x\to \infty$ . So,

  $\begin{array}{c}\underset{x\to -\infty }{\lim }f\left(x\right)=0\\ \underset{x\to \infty }{\lim }f\left(x\right)=\infty \end{array}$