The Exponential Function

Published: Dec 26th, 2020

Learning Objectives
By the end of this section, you should be able to:
1. Describe the properties of exponential functions
2. Graphically represent exponential functions

In this lesson, we will introduce the exponential function

Would you rather win $1,000,000 today or$0.01 that would double every day over 30 days? It may sound like a tough question right now, but let's think it through.

We can represent the doubling relationship using the exponential function$$\large{f(x) = ab^x +c}$$, where $$b \neq 1$$ (otherwise, it would just be a horizontal line!)

Let's dissect that formula, shall we?

Here's a neat demo of exponential functions. Use the sliders to observe the effect of changing the value of $$b$$ on the shape of the function. The $$b$$ slider covers $$[0,5]$$ while the $$a$$ slider covers $$[-5,5]$$. Observe what happens when $$0\lt b\lt1$$ and when $$b \gt 1$$

I'll give you some time to use the demo...

... still waiting...

Alright! Now that you've (hopefully) played around with the demo, here are some observations that you may have made (keeping $$a \gt 0$$):

• The function's domain is all real numbers, $$\mathbb{R}$$
• The function's range is all positive real numbers, $$\{y\in\mathbb{R}|y\gt0\}$$.
• The base $$b$$ is always positive $$b \gt 0$$
• Exponential Decay: when $$0\lt b\lt1$$, as $$x \rightarrow +\infty$$$$y \rightarrow 0$$; as $$x$$ approaches positive infinity, $$y$$ approaches zero! Conversely, as $$x \rightarrow -\infty$$$$y \rightarrow +\infty$$; as $$x$$ approaches negative infinity, $$y$$ approaches positive infinity.
• Exponential Growth: when $$b \gt 1$$$$x \rightarrow +\infty$$$$y \rightarrow \infty$$; as $$x$$ approaches positive infinity, $$y$$ also approaches positive infinity. Conversely, as $$x \rightarrow -\infty$$$$y \rightarrow 0$$; as $$x$$ approaches positive infinity, $$y$$ approaches zero.
• At $$x=0$$$$f(x) = a + c$$. This is the function's $$y$$-intercept (0, $$y$$).

Note that in both cases, we are only approaching zero and infinity. The fact that we can only approach—but never reach—zero implies that the function has an asymptote. More specifically, the basic exponential function has a horizontal asymptote at $$y=0$$.

Now, back to our question. Let's substitute the appropriate values into our exponential function, setting $$a=0.01$$, $$x=30$$, and $$b=2$$:

$$= (0.01)(2)^{30}$$

$$=10,737,418.24$$

Unless I'm wrong, ~11 million dollars is a much better deal than only \$1 million, no?

Exponential functions are powerful. While we will discuss them further in the next lesson, try some practice problems for now!