Calculus How To

Exponential Functions: Simple Definition, Examples

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Contents:

  1. Exponential Functions
  2. Nth Root Functions

What are Exponential Functions?

Exponential functions have the variable x in the power position.  For example, an exponential equation can be represented by:

f(x) = bx.

Like other algebraic equations, we are still trying to find an unknown value of variable x.


One way to think of exponential functions is to think about exponential growth—the idea of small increases followed by rapidly increasing ones.  These increases (or decreases when working with negative exponents) are consistent over a definite period of time as a function of the variable x. For example, the increases are consistently double or triple. The rapid increases characteristic of exponential functions can be seen on the graph below:

Calculus How To Exponential Functions

Most exponential graphs will have this same arc shape; There are some exceptions. For example, the graph of ex is nearly flat if you only look at the negative x-values:

exponential functions

Graph of ex. The graph of negative x-values (shown in red) is almost flat.


Exponential functions are an example of continuous functions

Graphing the Function

The base number in an exponential function will always be a positive number other than 1.  The first step will always be to evaluate an exponential function.  In other words, insert the equation’s given values for variable x and then simplify.

For example, we will take our exponential function from above, f(x) = bx, and use it to find table values for f(x) = 3x.

Step One: Create a table for x and f(x)


x f(x)

 

Step Two: Choose values for x.

x f(x)
-2
-1
0
1
2

 

Step Three: Evaluate the function for each value of x. “Evaluate” means to plug the x-values into the equation and solve.

x f(x)
-2 3-2 = 1/32  = 1/9
-1 1/3
0 1
1 3
2 9

The points in our chart are then plotted on the x-axis and y-axis of our graph giving us the following:

Calculus How To Graph Example Exponential Growth

Real World Uses

Exponential functions can also be applied in real world contexts to determine things like population growth and radioactive decay.  In these cases, the function will not be like above (f(x) = bx) but rather the formula must account for other factors.  For example, in the case of radioactive decay, the formula would look like this:

Calculus How To Exponential Functions Radioactive Decay

Where :

  • R = the remaining value of the substance
  • A = the initial amount of the substance (grams in the example)
  • h = the half-life of the substance
  • t = the amount of time passed (60 years in example)

Base Number

The term “base number” in calculus usually refers to the number found in exponential functions, which have the form

f(x) = ax

The “a” is the above expression is the base number, so-called because it’s a solid base(foundation) for the rest of the expression—like the hovering exponent (x) to the right. The base can be any positive real number not equal to 1.

A base number system is, like the name suggests, an entire number system based on a certain number of digits. For example, the decimal number system has 10 digits.

Logarithmic functions are also defined with respect to a “base”, which is any positive number. However, the “base” in a logarithm isn’t usually called a base number— just a “base”.

Contents:

  1. Bases in Exponential Functions
  2. Number Systems

1. Base Numbers (Exponents)

The base number also refers to the number that is multiplied by itself in an exponent. It’s usually written in a larger font.
The exponent tells us how many times to multiply the base number by itself. It’s usually written in a smaller font (as a superscript).
base number

In the above image, the base number is 8. In other words, multiply 8 by itself.  The exponent in this example is 3.  What this tells you is that you should multiply 8 by itself 3 times:

8 x 8 = 64 x 8 = 512

2. Number Systems

The term “Base number” also refers to the values in a number system.  More specifically, the base tells you how many digits are in the system. For example, the base 10 system has 10 digits from 0 to 9.
base number system

The number system with the fewest number of digits is the binary system, also known as a base-2 number system.  The two digits of the binary system are 0 and 1.  Each digit in the binary system is known as a bit.  The binary system is commonly used by most computer systems.

Natural Exponential Function

The natural exponential function, ex, is the inverse of the natural logarithm ln. The e in the natural exponential function is Euler’s number and is defined so that ln(e) = 1. This number is irrational, but we can approximate it as 2.71828.

It has one very special property: it is the one and only mathematical function that is equal to its own derivative (see: Derivative of ex). Looking at this graphically, this means that the slope of a tangent line to the curve at any point is equal to the height of the curve at that point.

The graph of ex is a reflection of ln(x) over the line y = x.

natural exponential function

The graph of ex (blue) is a reflection of ln(x) (red) over the line y = x (green).


The limit of ex as x goes to minus infinity is zero, and the limit as x goes to positive infinity is infinity.

These two functions are inverses of each other:
natural exponential function and inverse

Properties of the Natural Exponential Function

ex has some handy properties that make it very easy to manipulate algebraically. Because of the laws of exponents, we know that:

properties of natural exponential functions

Why Use the Natural Exponential Function

We don’t just use the natural exponential function because it makes algebra become easy, though. It’s also an accurate model of many processes in our world, including the growth and decay of biological systems and what happens to the money when it is compounded.

Nth Root Function

The nth root function is the inverse of the exponential function xn.

In simple terms, it does the opposite, or “undoes” the exponential. For example, if x = 2, the exponential function 2x would result in 22 = 4. The nth root (in this case, the cube root, √) takes the output (4), and gives the original input: √(4) = 2.

nth root function

The nth root function x(1 / 2)(blue) along with its inverse: the exponential function x2 (red).



The term “nth” just means an order: 1st, 2nd, 3rd, and so on. It is just shorthand notation for the entire group of endless possibilities.

Notation

The nth root function can be written in two ways:

n√· or, equivalently, x1 / n.

The second format—x1 / n— is what you’ll want to put into a graphing program or calculator. Some calculators do have nth root functions, like the Casio FX-9750G (press SHIFT and nth function casio 9750G).

Examples

The graph above shows an nth root function where n = 2. When n = 2, the function is called a square root function. It’s just a specific nth root with a specific name.

Another example of an nth root function is a cube root function, where n = 3:

3√(x).

For example, the cube root of -1 is -1: 3√(-1) = -1.

Formal Definition of an nth Root Function

The formal definition is:

n√· : [0, ∞] ℝ, given by
n√ (x) = the unique real number y ≥ 0 with yn = x.

The nth root function, n√(x) is defined for any positive integer n. However, there is an exception: if you’re working with imaginary numbers, you can use negative values. For example, (-1)½ = ± i, where i is an imaginary number.

Properties of the nth root Function

The nth root function is a continuous function if n is odd. If n is even, the function is continuous for every number ≥ 0. Note though, that if n is even and x is negative, then the result is a complex number.

References

Chapter 1 Review: Supplemental Instruction. Retrieved December 5, 2019 from: https://apps-dso.sws.iastate.edu/si/documentdb/spring_2012/MATH_165_Johnston_shawnkim_Chapter_1_Review_Sheet.pdf
Ellis, R. & Gulick, D. (1986). Calculus with Analytic Geometry. Harcourt Brace Jovanovich
Math 142a Winter 2014. Retrieved December 5, 2019 from: http://www.math.ucsd.edu/~drogalsk/142a-w14/142a-win14.html
Nau, R. The Logarithmic Transformation. Retrieved February 24, 2018 from: https://people.duke.edu/~rnau/411log.htm
Ving, Pheng Kim. Calculus of One Real Variable. Chapter 7: The Exponential and Logarithmic Functions. Retrieved from http://www.phengkimving.com/calc_of_one_real_var/07_the_exp_and_log_func/07_01_the_nat_exp_func.htm on July 31, 2019
Woodard, Mark. 7.3 The Natural Exp. Lecture Notes. Retrieved from http://math.furman.edu/~mwoodard/math151/docs/sec_7_3.pdf on July 31, 2019
Pilkington, Annette. Calculus 2 Lecture Slides. Lecture 3. Retrieved from https://www3.nd.edu/~apilking/Calculus2Resources/Lecture%203/Lecture_3_Slides.pdf

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Stephanie Glen. "Exponential Functions: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/types-of-functions/exponential-functions/
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