An exponential function is a mathematical function of the form:

$f(x) = a \cdot b^{x}$

Where:

• $a$ is a constant that represents the initial value (or the value of the function when $x = 0$),
• $b$ is the base of the exponential, and
• $x$ is the variable in the exponent.

Important Notes:

• The base $b$ is a positive real number, and $b \neq 1$.
• If $b > 1$, the function is exponentially increasing, meaning the value of $f(x)$ grows rapidly as $x$ increases.
• If $0 < b < 1$, the function is exponentially decreasing, meaning the value of $f(x)$ gets smaller as $x$ increases.

A special case of an exponential function is when $b = e$, where $e$ is Euler's number ($e \approx 2.718$). In this case, the function is called a natural exponential function, and it is written as:

$f(x) = a \cdot e^{x}$

Would you like an example of how to work with exponential functions or any specific part explained in more detail?

Here are some related questions:

1. How do you graph an exponential function?
2. What is the difference between exponential growth and exponential decay?
3. What is the natural exponential function, and how is it used in real life?
4. How do you solve equations involving exponential functions?
5. How do exponential functions relate to logarithmic functions?

Tip: When graphing exponential functions, always pay attention to the base $b$ and how it affects the steepness of the curve.