Exponential Functions

Lesson

An exponential function has the form $f(x) = a \cdot b^x$, where the variable sits in the exponent. The base $b$ is positive and not equal to 1. This shape appears anywhere a quantity is repeatedly multiplied — populations, money, radioactive decay.

Two key behaviors:

• Growth: when $b > 1$, the function increases as $x$ grows.
• Decay: when $0 < b < 1$, the function decreases.

Either way, the graph never touches the $x$-axis: the horizontal asymptote is $y = 0$. Domain is all real numbers; range is $(0,\,\infty)$ when $a > 0$. And $f(0) = a \cdot b^0 = a$, so the $y$-intercept is just $a$.

To evaluate, substitute the input and use exponent rules:

• $b^0 = 1$.
• $b^{-n} = \dfrac{1}{b^n}$.
• $b^{1/n} = \sqrt[n]{b}$.
• $b^{m/n} = (\sqrt[n]{b})^m$.