13.1. Exponential functions

An exponential function is a function containing terms with the form b^x. The number b is called the base; it may be any positive real number such that b ≠ 1. The exponent x can be any real number. Restricting the range of the base b to positive number avoids complex valued exponential terms. For example:

(-4)^0.5 = 2i

0x = 0 for x ≠ 0
1x = 1

Properties of exponential functions

For the exponential function: f(x) = b^x

The domain of f(x) is: (-∞, +∞); its range is: (0, +∞)

The function is continuous, increasing, and is one to one.

A function must have one value of y for each value of x. A one to one function has one value of x for each value of y. This condition must be met for a function to have an inverse.

The graph of f(x) is concave upward.

b⁰ = 1

b^x > 0 for all x

If b^x = b^y then x = y

The graph y = (1/b)^x is symmetric to the graph of y = b^x about the y-axis.

If b > 1, then

b^x will increase without bound as x -> +∞

lim bx = 0
x -> -∞

If b < 1, then

b^x will increase without bound as x -> -∞

lim bx = 0
x -> +∞

For example, graph: f(x) = e^x

 x      ex
----------------
-4      0.018316
-3      0.049787
-2      0.135335
-1      0.367879
 0      1.000000
 1      2.718282
 2      7.389056
 3     20.085537
 4     54.598150

Figure 13-1: f(x) = e^x