Exponential function

Definition

An exponential function is a mathematical expression 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 of an exponential function determines whether it represents growth ($b > 1$) or decay ($0 < b < 1$).
2. The y-intercept of an exponential function occurs at $(0, a)$ since $f(0) = a \cdot b^0 = a$.
3. Exponential functions have a horizontal asymptote, usually the x-axis (y=0), unless shifted vertically.
4. The rate of change in an exponential function increases/decreases multiplicatively, unlike linear functions which change additively.
5. In solving exponential equations, logarithms are often used to isolate the variable in the exponent.

Review Questions

• What are the key characteristics that differentiate an exponential function from other types of functions?
• How do you determine if an exponential function represents growth or decay?
• Why are logarithms useful when working with exponential functions?

Related terms

Logarithmic Function: A logarithmic function is the inverse of an exponential function and can be written as $y = \log_b{x}$ where $b$ is the base.

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

Growth Factor: $b$ in the context of $f(x) = a \cdot b^x$, which determines how quickly the value grows or decays.