Chapter 4.3, Problem 32E

Solution

The given function $f\left(x\right)=\frac{1}{3}{e}^{4x}$ is exponential growth or not.

The given function $f\left(x\right)=\frac{1}{3}{e}^{4x}$ is an exponential growth.

Given information: $f\left(x\right)=\frac{1}{3}{e}^{4x}$

Calculation:

The function is,

  $f\left(x\right)=\frac{1}{3}{e}^{4x}$

Let $f\left(x\right)$ = $y$

Therefore,

  $f\left(x\right)=\frac{1}{3}{e}^{4x}$ (1)

A function $y$ = $a{b}^x$ is exponential decay when $b$ falls between 0 and 1. The value of $a$ can never be zero and the value of b can never be one.

A function $y$ = $a{b}^x$ is exponential growth when $b$ is greater than one. The value of $a$ can never be zero.

Now put the values like 0,1, 2,3 in (1)

Column1	Column2	Column3	Column4
x	0	1	2
y	0.3333	18.1993	993.6526

  

Note that has $x$ become larger the value of $e^{4x}$ grows without bound. So, this is an exponential growth function.

From the graph, it can be seen as the values increase every time.

Therefore, the given function is an exponential growth function.