Explosive growth or rapid decay — the function behind compound interest, population, and radioactivity.
An exponential function has the variable x in the exponent position, written as f(x) = a·bˣ, where the base b is a positive constant not equal to 1, and a ≠ 0. When b > 1, the function grows exponentially — it starts slow and then increases dramatically (exponential growth).
When 0 < b < 1, the function decays — it decreases rapidly toward zero (exponential decay). 718 as its base: f(x) = eˣ.
Exponential functions are essential for modeling any process where growth or decay is proportional to the current amount.
f(x) = a · bˣ
Initial value — the y-intercept (value when x = 0)
Base — the growth/decay factor (b > 0, b ≠ 1)
Exponent — the input variable (the key feature of exponential functions)
Domain
All real numbers: (-∞, +∞)
Range
(0, +∞) when a > 0 (never reaches zero)
Y-intercept
(0, a)
Horizontal Asymptote
y = 0 (the x-axis)
Growth (b > 1)
Increasing; rises steeply as x increases
Decay (0 < b < 1)
Decreasing; approaches but never reaches zero
f(x) = 2ˣClassic exponential growth: doubles with each unit increase in x.
f(x) = eˣNatural exponential: continuous growth rate, base e ≈ 2.718.
f(x) = (0.5)ˣExponential decay: halves with each unit increase in x.
Change the base and watch growth vs. decay shift in real time
In a polynomial like x², the base is the variable and the exponent is constant. In an exponential like 2ˣ, the base is constant and the exponent is the variable. Exponential functions grow (or decay) far faster than any polynomial for large x.