Exponential Functions

In a linear function you add the same amount each step. In an exponential function you *multiply* by the same amount each step.

y = a·bˣ, where a is the starting value and b is the multiplier per step.

If b > 1 the function grows. If 0 < b < 1 it decays.

y = 3·2ˣ starts at 3 and doubles each time: 3, 6, 12, 24, 48.

The difference matters enormously over time. Adding 5 each step for 20 steps gets you to 100. Doubling for 20 steps gets you past a million.

The graph never touches the x-axis. It gets closer and closer, which makes the x-axis a horizontal asymptote.

An exponential function f(x) = a·bˣ with a ≠ 0, b > 0 and b ≠ 1 has domain ℝ, range (0, ∞) for a > 0, and y-intercept (0, a) since b⁰ = 1. The line y = 0 is a horizontal asymptote.

Its defining property is a constant multiplicative rate: f(x+1)/f(x) = b for all x, in contrast to a linear function's constant difference. Tables are classified by exactly this test — constant differences mean linear, constant ratios mean exponential.

The base is restricted to positive values because bˣ is not real-valued for negative b at fractional exponents; (−4)^{1/2} has no real value.

Exponential growth eventually dominates any polynomial: for b > 1, bˣ/xⁿ → ∞ as x → ∞ for every fixed n. That fact is why compound interest, population models and algorithmic complexity are all discussed in exponential terms.

Inverting an exponential requires the logarithm, which is the central subject of Algebra 2.