Lesson 3: Exponential 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

Properties of the Graph of $f(x) = a^x$

Examples

• The graph of $f(x) = 4^x$ is an increasing curve because $a > 1$. It passes through the points $(0, 1)$ and $(1, 4)$.
• The graph of $g(x) = (\frac{1}{4})^x$ is a decreasing curve because $0 < a < 1$. It passes through the points $(0, 1)$ and $(-1, 4)$.
• Both graphs $f(x) = 4^x$ and $g(x) = (\frac{1}{4})^x$ have a domain of all real numbers, $(-\infty, \infty)$, and a range of all positive numbers, $(0, \infty)$.