Exponential function

Definition

An exponential function is a mathematical function in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.

5 Must Know Facts For Your Next Test

1. The base $b$ determines whether the function represents exponential growth ($b > 1$) or decay ($0 < b < 1$).
2. The graph of an exponential function is always a smooth curve that either increases or decreases rapidly.
3. The y-intercept of an exponential function $f(x) = a \cdot b^x$ occurs at $(0, a)$.
4. Asymptotes for exponential functions are horizontal lines that the graph approaches but never touches; typically, this is the x-axis (y=0).
5. Exponential functions have continuous domains (all real numbers), but their ranges depend on the sign of $a$: if $a > 0$, range is $(0, \infty)$; if $a < 0$, range is $(-\infty, 0)$.

Review Questions

• What does the base of an exponential function tell you about its behavior?
• How do you find the y-intercept of an exponential function?
• What kind of asymptote does an exponential function have?

Related terms

Logarithmic Function: The inverse of an exponential function, expressed as $f(x) = \log_b(x)$ where $b$ is the base.

Asymptote: A line that a graph approaches but never touches. Exponential functions often have horizontal asymptotes.

Growth Factor: In an exponential growth function, it is the base ($b$) when it’s greater than one.