Introduction to Exponential Functions
Introduction
Exponential functions play a fundamental role in mathematics and are widely used in various fields, including science, finance, and computer science. We will explore the basics of exponential functions, their properties, and how to work with them.
A simple example
When you have a binary tree structure, the number of leafs of the tree can be calculated with the function
$L(l)=2^l$ where $l$ is the number of levels of the tree.
Another example
If you are interested in learning what is the rate of growth of money deposited (loaned) with an interest rate of $r$ compounded annually then you can figure that out using the exponential function
$R(y)= (1+r)^y$ where $y$ is the number of years the money would be in the deposit.
What is an Exponential Function?
An exponential function is a mathematical function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is a variable. The base $a$ is typically greater than 1, but it can also be a number between 0 and 1, excluding 0. The variable $x$ can be any real number or even a complex number. Even though many of the facts that we will expose also hold for exponentials with complex number values we will mainly focus on the variable taking real inputs.
Properties of Exponential Functions
Growth or Decay
Exponential functions can represent both growth and decay phenomena. When the base $a$ is greater than 1, the function exhibits exponential growth. Conversely, when 'a' is between 0 and 1, excluding 0, the function exhibits exponential decay.
Domain and Range
The domain of an exponential function is the set of all real numbers. The range depends on whether the function base is or is not 1. If the base of the exponential function is different than 1, then the range is all positive real numbers. If the base of the exponential function is 1, then the range is just the set ${1}$.
Continuous Growth
Exponential functions exhibit continuous growth or decay, meaning that they change gradually over time rather than in discrete steps.
Asymptote
Exponential functions have a horizontal asymptote, which is a line that the function approaches but never reaches. The asymptote is typically $y = 0$ for growth functions when approching $-\infty$ and $y = 0$ for decay functions when approching $(+)\infty$.
Additive property of Exponents
Probably the most important property of exponential functions, and in fact one of its defining characteristics, is that it opens up sums into products. Namely
- $f(x + y) = f(x) f(y) $
This property bears a lot of interesting facts about exponentials, including some of the following:
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$f(0) = 1$ or $f(0) = 0$ but we will not deal with the $f(0)=0$ case since:
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If $f(0) = 0$ then $f(x)= 0$ for all $x$
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$f(-x) = \frac{1}{f(x)}$
Common Applications:
Exponential functions have numerous applications in various fields. Some common applications include:
- Population growth and decay
- Compound interest and investments
- Radioactive decay
- Biological processes
- Epidemic modeling
- Electronics and signal processing