# 4.1 Exponential functions  (Page 2/16)

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Apparently, the difference between “the same percentage” and “the same amount” is quite significant. For exponential growth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding 2 to the output whenever the input was increased by one.

The general form of the exponential function is $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is any nonzero number, $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is a positive real number not equal to 1.

• If $\text{\hspace{0.17em}}b>1,$ the function grows at a rate proportional to its size.
• If $\text{\hspace{0.17em}}0 the function decays at a rate proportional to its size.

Let’s look at the function $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ from our example. We will create a table ( [link] ) to determine the corresponding outputs over an interval in the domain from $\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}3.$

 $x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $f\left(x\right)={2}^{x}$ ${2}^{-3}=\frac{1}{8}$ ${2}^{-2}=\frac{1}{4}$ ${2}^{-1}=\frac{1}{2}$ ${2}^{0}=1$ ${2}^{1}=2$ ${2}^{2}=4$ ${2}^{3}=8$

Let us examine the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ by plotting the ordered pairs we observe on the table in [link] , and then make a few observations.

Let’s define the behavior of the graph of the exponential function $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ and highlight some its key characteristics.

• the domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$
• the range is $\text{\hspace{0.17em}}\left(0,\infty \right),$
• as $\text{\hspace{0.17em}}x\to \infty ,f\left(x\right)\to \infty ,$
• as $\text{\hspace{0.17em}}x\to -\infty ,f\left(x\right)\to 0,$
• $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is always increasing,
• the graph of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ will never touch the x -axis because base two raised to any exponent never has the result of zero.
• $\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$ is the horizontal asymptote.
• the y -intercept is 1.

## Exponential function

For any real number $\text{\hspace{0.17em}}x,$ an exponential function is a function with the form

$f\left(x\right)=a{b}^{x}$

where

• $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the a non-zero real number called the initial value and
• $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is any positive real number such that $\text{\hspace{0.17em}}b\ne 1.$
• The domain of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is all real numbers.
• The range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is all positive real numbers if $\text{\hspace{0.17em}}a>0.$
• The range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is all negative real numbers if $\text{\hspace{0.17em}}a<0.$
• The y -intercept is $\text{\hspace{0.17em}}\left(0,a\right),$ and the horizontal asymptote is $\text{\hspace{0.17em}}y=0.$

## Identifying exponential functions

Which of the following equations are not exponential functions?

• $f\left(x\right)={4}^{3\left(x-2\right)}$
• $g\left(x\right)={x}^{3}$
• $h\left(x\right)={\left(\frac{1}{3}\right)}^{x}$
• $j\left(x\right)={\left(-2\right)}^{x}$

By definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus, $\text{\hspace{0.17em}}g\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ does not represent an exponential function because the base is an independent variable. In fact, $\text{\hspace{0.17em}}g\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ is a power function.

Recall that the base b of an exponential function is always a positive constant, and $\text{\hspace{0.17em}}b\ne 1.\text{\hspace{0.17em}}$ Thus, $\text{\hspace{0.17em}}j\left(x\right)={\left(-2\right)}^{x}\text{\hspace{0.17em}}$ does not represent an exponential function because the base, $\text{\hspace{0.17em}}-2,$ is less than $\text{\hspace{0.17em}}0.$

Which of the following equations represent exponential functions?

• $f\left(x\right)=2{x}^{2}-3x+1$
• $g\left(x\right)={0.875}^{x}$
• $h\left(x\right)=1.75x+2$
• $j\left(x\right)={1095.6}^{-2x}$

$g\left(x\right)={0.875}^{x}\text{\hspace{0.17em}}$ and $j\left(x\right)={1095.6}^{-2x}\text{\hspace{0.17em}}$ represent exponential functions.

## Evaluating exponential functions

Recall that the base of an exponential function must be a positive real number other than $\text{\hspace{0.17em}}1.$ Why do we limit the base $b\text{\hspace{0.17em}}$ to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:

• Let $\text{\hspace{0.17em}}b=-9\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=\frac{1}{2}.\text{\hspace{0.17em}}$ Then $\text{\hspace{0.17em}}f\left(x\right)=f\left(\frac{1}{2}\right)={\left(-9\right)}^{\frac{1}{2}}=\sqrt{-9},$ which is not a real number.

Why do we limit the base to positive values other than $1?$ Because base $1\text{\hspace{0.17em}}$ results in the constant function. Observe what happens if the base is $1:$

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
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