Exponential Functions

An exponential function is of the form f( x) = a, for some real number a, as long as a > 0. While exponential functions accept any real number input for x, the range is limited to positive numbers.

Natural exponential function

Although you will deal with many, the most common exponential function you'll encounter is the natural exponential function, written as f( x) = e x. Although the base e looks just as generic as the base a in our definition of exponential function, it is not. The e stands for Euler's number, and represents a standard, commonly known, irrational constant, sort of like π. (Note that “Euler” is pronounced “OIL‐er,” not “YOU‐ler.”)



Although the decimal digits in e appear to repeat themselves at first (2.718281828…), they soon diverge into an non‐repeating and non‐terminating pattern.

You are not expected to memorize e, just as you are not expected to memorize π therefore, answers can be left in terms of e (such as 12 e 5) and are still considered simplified. If you are expected to evaluate the natural exponential function, you will be allowed to use a calculator.



All scientific and graphing calculators have an e button, but be aware that in some tools and graphing software packages, the e x button is labeled as “exp.”

Graphs of exponential functions

Consider the graph of f( x) = 2 x in Figure , plotted by substituting a small collection of integers into f.



Figure 1 The graph of fx) = 2 x .

From the table, it is clear why the graph of f gets closer and closer to the x‐axis as x gets more and more negative. Since a negative input becomes a negative exponent, your result will be a fraction. The larger the negative input, the smaller the value of the fraction. However, once the inputs become positive, the graph grows quickly.



One other thing becomes clear upon examination of f( x). Note that f(0) = 1. That is not true only when the base of the exponential function is 2. In fact, no matter what the base a in g( x) = a x , the graph of g will contain the point (0,1) since any positive number a raised to the zero power will be 1.

Since you can be sure the point (0,1) will appear in an exponential graph, use it as the anchor point for your sketch if asked to transform the graph of an exponential function.

Example 1: Sketch the graph of h( x) = −3 x + 2 − 1.

Your graph should be y = 3 x reflected across the x‐axis, moved two units to the left, and one unit down. Think about this in terms of the anchor point (0,1). The initial reflection changes the point to (0, −1); once shifted left and down, the anchor point ends up at (−2, −2). The former horizontal asymptote of y = 0 has also moved down one, so the new horizontal asymptote is y = −1. See Figure .

Figure 2 The graph of y = 3 x, shown as a dotted curve, and the graph of h( x), the answer to Example 1.