Evaluating Exponential Functions

Learning Outcomes

  • Identify and evaluate exponential functions

A function of the form [latex]f(x)=b^{x}, b>0, b \neq 1[/latex] is called an exponential function. In an exponential function the base is a positive constant other than [latex]1[/latex], and the exponent is a variable expression. For example, if [latex]f(x)=3^{x}[/latex], then [latex]f(2)=3^{2}=9, f(0)=3^{0}=1[/latex] and [latex]f(-2)=3^{-2}=\frac{1}{3^{2}}=\frac{1}{9}[/latex].

Exponential Growth function

A function that models exponential growth grows by a rate proportional to the current amount. For any real number x and any positive real numbers a and b such that [latex]b\ne 1[/latex], an exponential growth function has the form

[latex]f\left(x\right)=a{b}^{x}[/latex]

where

  • a is the initial or starting value of the function.
  • b is the growth factor or growth multiplier per unit x

Evaluating Exponential Functions

To evaluate an exponential function of the form [latex]f\left(x\right)=a \cdot {b}^{x}[/latex], we simply substitute x with the given value, and calculate the resulting power. For example:

Let [latex]f(x)=7 \cdot 2^{x}[/latex]

Then [latex]2[/latex] is the base of the exponential function. Since [latex]f(x)=7 \cdot 2^{0} = 7 \cdot 1= 7[/latex], the constant [latex]a=7[/latex] in this example is the initial value of the function when the variable [latex]x[/latex] is [latex]0[/latex].

If [latex]x[/latex] is [latex]1[/latex], then
[latex]f(1)=7 \cdot 2^{1}[/latex]
[latex]= 7 \cdot 2[/latex]
[latex]= 14[/latex].

If [latex]x[/latex] is [latex]3[/latex], then
[latex]f(3)=7 \cdot 2^{3}[/latex]
[latex]= 7 \cdot 8[/latex]
[latex]= 56[/latex].

To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. Consider the following example.

Example

Let [latex]f\left(x\right)=5{\left(3\right)}^{x+1}[/latex]. Evaluate [latex]f\left(2\right)[/latex] without using a calculator.

In the following video, we present more examples of evaluating an exponential function at several different values.

A General Note: The Number [latex]e[/latex]

The letter e represents the irrational number

[latex]{\left(1+\frac{1}{n}\right)}^{n}[/latex]
as n increases without bound

The letter e is used as a base for many real-world exponential models. To work with base e, we use the approximation, [latex]e\approx 2.718282[/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.

The Continuous Growth/Decay Formula

For all real numbers r, t, and all positive numbers a, continuous growth or decay is represented by the formula

[latex]A\left(t\right)=a{e}^{rt}[/latex]

where

  • a is the initial value,
  • r is the continuous growth or decay rate per unit time,
  • and t is the elapsed time.

If r >[latex]0[/latex], then the formula represents continuous growth. If r < [latex]0[/latex], then the formula represents continuous decay.

 

For business applications, the continuous growth formula is called the continuous compounding formula and takes the form

[latex]A\left(t\right)=P{e}^{rt}[/latex]

where

  • P is the principal or the initial amount invested,
  • r is the growth or interest rate per unit time,
  • and t is the period or term of the investment.

Example

The population of a town, in thousands, [latex]t[/latex] years after [latex]2013[/latex] is modeled by the function [latex]A(t)=13.2 \cdot e^{0.07t}[/latex]. Find the population of the town in the year…

  1. 2013
  2. 2020

In the following video, we show an example of calculating the remaining amount of a radioactive substance after it decays for a length of time.