Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.6 Radical Functions
Forms
Generally, an exponential function is defined as repeated multiplication of some number. A most basic form could be f(x)= bx where b>0, b≠1. Here, b stands for the base of the exponential, or the number that is being repeatedly multiplied. Why should we restrict the base?
There are a few cases:
b>1: If the base is larger than one then we see an exponential growth function. Multiplying by something bigger than one makes a number bigger. So, the functions value grows as the input grows.
b=1: This is not an exponential function. If we multiply by one repeatedly, the number will never change. So, a function whose base is one is a horizontal, linear function. Technically, there are no issues with this, but it is better classified as linear than exponential.
0<b<1: Here we have bases that are fractional. That is to say, when we repeatedly multiply, we are making the value smaller. Functions with a base smaller than one are called exponential decay functions.
b=0: If we take the base to be zero it works… half of the time. If the exponent is greater than 0, we get 0. However, if we take the exponent to be 0 or negative, we have some pretty serious issues. 00 is indeterminate and 0(negative number) is undefined. A function with a 0 base cannot exist because most of the values don’t exist and the rest of them are just 0!
b<0: Here we have an extremely interesting case. Let’s take an example to help us. If b=-1, then we have y=(-1)x. If we start plugging in so that we can evaluate this function we get values like the following:
x=0,y=0; x=1,y=-1; x=2,y=1
But we also get:
x=0.5, y= 
A negative value for the base results in imaginary outputs! This gets progressively more difficult too: what would happen if we plugged in x=0.25? This is the reason that the base is restricted to positive numbers only for an exponential function.
Graph
There are two main graphs for an exponential function.
Growth: f(x)= bx,b>1
Decay f(x)= bx,0<b<1
Example 1: Graph y=1.23x+2
Domain and Range
The domain of an exponential function has no restrictions. It is the set of all real numbers.
As you can tell from the graphs, the range is restricted to positive numbers only. The reason for this is because of the restriction of the base. If the base is a positive number, then raising it to any exponent will result in a positive multiplied by a positive some number of times. This will always result in a positive number. Another way to think of this is that it is impossible to multiply a positive by itself and get a negative.
Special Forms
Using a constant percent change, like with interest rates and finance, we can write an equation to calculate the amount after a certain amount of time periods: A(t)=a(1+r)t
In this formula, a is the initial value, r is the percent change, t is the number of time periods, and A(t) represents the output after t time periods. If r>0 then it is a growth function, if r<0 it is a decay function.
Another form that is helpful is the compound interest formula:
Here, we apply the interest rate, r, n times during a year.
A general form for exponential equations is often represented as P=P0bx
Where P0 is the initial value and P is the output. Sometimes b is written as ek. This is especially helpful for population calculations.
Example 2: Write the equation for the amount of money in an account if it compounds monthly with an interest rate of 4.5% and begins with $130.
Solution:
Applications
There are a wide range of applications for exponential functions. One of the most common is population calculations. This is because many populations have a growth rate dependent on the number in the population.
Example 3: Find the population of cells in an environment after an hour if they double every 10 minutes, beginning with 4 cells.
Solution: An equation that represents this could be P=4(2)(t⁄10). As we will discuss later, this isn’t ideal, but finding a better equation takes more work that we aren’t yet ready for. However, for this section the equation will function just fine. So, we can set t=60 because 1 hour is 60 minutes.
P=4(2)(60⁄10)
P=4∙26
Thus, after an hour, there will be P=256 cells.
Another example could be how things change with respect to a previous value, like a bouncing ball.
Example 4: Assume a ball is dropped from 12 feet. If each bounce is 4/5 the height of the previous bounce, how high will it reach after 4 bounces?
Solution: H(b)=12(4/5)b where H(b) is the height of the ball after b bounces.
Growth is also used to help model interest rates and values of objects.
Example 5: Calculate the value of $100 invested at a 3% interest rate for 10 years, compounded monthly.
Solution:
Review
An Exponential Functions is a function where a number is being repeatedly multiplied by itself, depending on the input.
The base, b, should be positive.
There are two types. Growth: f(x)= bx,b>1 and Decay: f(x)= bx, 0<b<1
Equations
General Form: f(x)= y0 bax where b>0,b≠1
Compound Interest P= P0 (1+r/n)nt