Chapter 6.1, Problem 46E

a.

To determine

Whether the graph of $f$ represent exponential growth or exponential decay.

Answer to Problem 46E

  $\therefore f\left(x\right)$ represents an exponential decay function.

Explanation of Solution

Given information:

  $f\left(x\right)$ is exponential function is given by $f\left(x\right)=a{b}^x$

  

Concept used:

An exponential function has the form $y=a{b}^x$ where $a\ne 0$ and $b$ is positive real number other than $1$ .

For exponential growth: $b\rangle 1$

For exponential decay: $0\langle b\langle 1$

Calculation:

$x$$-1$$0$$1$$2$$f\left(x\right)$$4$$1$$\frac{1}{4}$$\frac{1}{16}$

  $f\left(x\right)=a{b}^x$

  $\begin{array}{l}a=1\left[\text{since }f\left(a\right)\text{=1}\right]\\ b=\frac{\frac{1}{4}}{1}=\frac{1}{4}\\ \therefore f\left(x\right)=1{\left(\frac{1}{4}\right)}^x\\ Now,\\ 0\langle \frac{1}{4}\langle 1\end{array}$

  $\therefore f\left(x\right)$ represents an exponential decay function.

b.

To determine

To find: The domain and range of the function.

Answer to Problem 46E

The domain of the function is all real number and the range of the function is $\left(0,\infty \right)$ .

Explanation of Solution

Given information:

  

  $f\left(x\right)=1\propto {\left(\frac{1}{4}\right)}^x$

Concept used:

Domain is the set of all possible input (x-values) values.

Range is the set of all possible input (y-values) values.

Calculation:

  

The x-axis is an asymptote of the graph.

So, the domain of the function is all real numbers.

From the graph, it is clear that $f\left(x\right)\rangle 0$ for all values of $x$ .

So, the range of the function is $\left(0,\infty \right)$