Lesson 1: Exponential Growth and Decay Functions

Property

An exponential function is a function of the form $f(x) = b^x$, where $b$ is a positive real number $(b > 0)$ and $b \neq 1$. The domain of an exponential function is the set of all real numbers.

Examples

• $f(x) = 5^x$ is an exponential function since the base, 5, is a positive number not equal to 1.

• $g(x) = (\frac{1}{3})^x$ is an exponential function because its base, $\frac{1}{3}$, is positive and not equal to 1.

Property

For an exponential function $f(x) = b^x$:

• The graph always passes through the point $(0, 1)$, which is the y-intercept.
• The x-axis ($y=0$) is a horizontal asymptote, meaning the graph gets infinitely close but never touches it.
• If the base $b > 1$, the function is always increasing (representing exponential growth).
• If $0 < b < 1$, the function is always decreasing (representing exponential decay).

Examples

• The graph of $f(x) = 3^x$ is an increasing function. It passes through $(0, 1)$ and rises sharply to the right as $x$ increases.