Chapter 7.6, Problem 38PPE

To determine

To evaluate: The given exponential function over a given domain and state if the values of range increase or decrease as domain increases.

Answer to Problem 38PPE

Rangeover given domain: $\left\{\frac{4}{100},\frac{4}{10},4,40,400,4000\right\}$

Range increases as domain increases.

Explanation of Solution

Given information:

Domain: $\left\{-2,-1,0,1,2,3\right\}$

Exponential function: $g\left(x\right)=4\cdot {10}^x$

Concept Used:

For a given function $f\left(x\right)$ represented by y , domain is all possible values that x can take and range is all possible values of $f\left(x\right)$ or y .

Calculation:

To evaluate the function $g\left(x\right)=4\cdot {10}^x$ over the given domain, take x values from domain one at a time and evaluate y for one.

For x = −2

  $\begin{array}{c}g\left(x\right)=4\cdot {10}^{-2}\\ =4\cdot \frac{1}{10\times 10}\\ =4\cdot \frac{1}{100}\\ =\frac{4}{100}\end{array}$

For x = −1

  $\begin{array}{c}g\left(x\right)=4\cdot {10}^{-1}\\ =\frac{4}{10}\end{array}$

For x = 0

  $\begin{array}{c}g\left(x\right)=4\cdot {10}^0\\ =4\end{array}$

For x = 1

  $\begin{array}{c}g\left(x\right)=4\cdot {10}^1\\ =40\end{array}$

For x = 2

  $\begin{array}{c}g\left(x\right)=4\cdot {10}^2\\ =4\times 100\\ =400\end{array}$

For x = 3

  $\begin{array}{c}g\left(x\right)=4\cdot {10}^3\\ =4\times 1000\\ =4000\end{array}$

Thus Range is $\left\{\frac{4}{100},\frac{4}{10},4,40,400,4000\right\}$ .

Hence range increases as domain increases.