Exponential Functions

Definition

An exponential function has the form: $f(x) = a \cdot b^x$

where:

• $a$ = initial value (y-intercept when $x = 0$)
• $b$ = base (growth/decay factor)
• $x$ = exponent (input variable)

Growth vs. Decay

Exponential Growth: $b > 1$

• Function increases
• Example: $f(x) = 2^x$

Exponential Decay: $0 < b < 1$

• Function decreases
• Example: $f(x) = (\frac{1}{2})^x$

Properties

• Domain: All real numbers
• Range: $(0, \infty)$ if $a > 0$
• Y-intercept: $(0, a)$
• Horizontal asymptote: $y = 0$
• Never touches or crosses x-axis

Exponential Growth/Decay Formula

$A = A_0(1 + r)^t$

where:

• $A$ = final amount
• $A_0$ = initial amount
• $r$ = rate (as decimal)
• $t$ = time

Growth: $r > 0$ (add) Decay: $r < 0$ (subtract)