Exponential Function

Property An exponential function is a function of the form $f(x) = ab^x$, where $a$ and $b$ are nonzero constants and $b$ is a positive number not equal to 1. Explanation Think of an exponential function as a 'super multiplier' machine! The variable $x$ is in the exponent, which means as $x$ changes, the $y$ values are repeatedly multiplied by the base $b$. This creates either rapid growth or decay, making the function shoot up or dive down incredibly fast. Examples Evaluate $f(x) = 3^x$ for $x = 2$. Solution: $f( 2) = 3^{ 2} = \frac{1}{3^2} = \frac{1}{9}$. Evaluate $f(x) = 4(2)^x$ for $x = 3$. Solution: $f(3) = 4(2)^3 = 4(8) = 32$. The population of a town is modeled by $P(t) = 5000(1.03)^t$, where $t$ is the number of years.