What Is an Exponential Function?

(Video) An Introduction to Exponential Functions

by ThinkwellVids

Here's a great video by ThinkwellVids. He very clearly explains what exponential functions are and shows some examples.

$\large{\text{Summary}}$

An exponential function is one where there's a constant being raised to a variable. The constant has to be positive and cannot equal 1. So for example, $4^x$ is an exponential function.

Exponential Function

An exponential function is a mathematical function in the form: $$f(x) = a ^ x$$ Here $a$ is a positive, real number (called the base) and $x$ is the input (independent variable). The thing you need to note about exponential functions is the fact that the independent variable is always in the exponent. This could by $x$, $3x$ or $15x^3$. The exponential function is an important function in math!

Typically, exponential functions are used in growth and decay problems. Check out the hyperlinked lessons for explanations on how a function grows and decays.

Rules of Exponential Functions

There are some tips and tricks when dealing with exponential functions.

If you have $x = 0$, then $a^0$ is always going to be $1$, no matter what $a$ is.

If you have $a^{\frac{x}{y}}$, it can be rewritten as $\sqrt[y]{a^x} = (\sqrt[y]{a})^x$

If you have $a^{-x}$, it can be rewritten as $\frac{1}{a^x}$

If you have $a^x \times a^y$, it can be rewritten as $a^{x+y}$

Similarly, if you have $\frac{a^x}{a^y}$, it can be rewritten as $a^{x-y}$

Graphs of Exponential Functions

Since an exponential function can only have a positive base, you might think that all of the graphs are going to look positive. Think again. We are going to think of the graphs in terms of $a \geq 1$ and $0 < a < 1$.

The following is a graph of $a \geq 1$, specifically $a = 2$

Notice how as $x$ increases, $y$ also increases, but at a more rapid pace. The graph of other numbers $a \geq 1$ (greater than zero) will also follow a similar construct.

Now let's look at when $0 < a < 1$, specifically when $a = 0.5$

Notice how as $x$ decreases and becomes more negative, $y$ increases at a rapid pace, similarly to above. The graph of other numbers between $0$ and $1$ will follow a similar construct.

You can also extend this idea to the exponential form of a complex number! Furthermore, continuously compounded interest and compound interest problems are real-world applicationa of exponential functions.

(Video) Exponential Function

by mathman1024

Here's a video by mathman1024 going over exponential functions.

$\large{\text{Summary}}$

Exponential function with base $a$: $$\text{If }a>0\text{, }a\ne1\\ f(x)=a^x$$

Some properties about exponential functions:

1. $a^x$ is unique for all $x$ (one-to-one)
2. $a^b=a^c\leftrightarrow b=c$
3. If $a>1$, $f(x)=a^x$ is increasing
4. If $0<a<1$, $f(x)=a^x$ is decreasing

Exponential Functions

An exponential function is a function in which a constant base is raised to a variable exponent. Unlike linear functions, which change by adding the same amount each time, exponential functions change by multiplying by the same factor repeatedly.

You may have seen exponential functions used to model many real-world problems, like population growth, compound interest, and radioactive decay.

####The General Form

An exponential function is usually written as:

$$f(x)=ab^x$$

, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent.

In order for this function to work, there are some rules:

• $b$ must be positive
• $b \neq 1$
• the variable must appear in the exponent

For example, $y=2^x$ is an exponential function because the variable $x$ is in the exponent.