Functions

Definition

A function $f$ from set $A$ to set $B$ is a rule that assigns to each element $x$ in $A$ exactly one element $y$ in $B$.

$f: A \to B$

$y = f(x)$

• Domain: Set of all possible input values ($x$)
• Range: Set of all possible output values ($y$)
• Codomain: The set $B$ (may include values not in range)

Types of Functions

Constant Function

$f(x) = c \quad \text{(where } c \text{ is a constant)}$

• Domain: All real numbers $\mathbb{R}$
• Range: $\{c\}$
• Graph: Horizontal line

Linear Function

$f(x) = mx + b$

• Domain: $\mathbb{R}$
• Range: $\mathbb{R}$
• Graph: Straight line with slope $m$, $y$-intercept $b$

Quadratic Function

$f(x) = ax^2 + bx + c \quad (a \neq 0)$

• Domain: $\mathbb{R}$
• Range: $[k, \infty)$ if $a > 0$, or $(-\infty, k]$ if $a < 0$
• Vertex: $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$
• Graph: Parabola

Polynomial Function

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

• Domain: $\mathbb{R}$
• Degree $n$ determines end behavior
• At most $n$ real zeros
• At most $n-1$ turning points

Rational Function

$f(x) = \frac{P(x)}{Q(x)} \quad \text{where } P \text{ and } Q \text{ are polynomials}$

• Domain: All $x$ where $Q(x) \neq 0$
• Vertical asymptotes where $Q(x) = 0$
• Horizontal/oblique asymptotes determined by degrees

Exponential Function

$f(x) = a^x \quad (a > 0, a \neq 1)$

• Domain: $\mathbb{R}$
• Range: $(0, \infty)$
• $y$-intercept: $(0, 1)$
• Horizontal asymptote: $y = 0$
• If $a > 1$: increasing (growth)
• If $0 < a < 1$: decreasing (decay)

Logarithmic Function

$f(x) = \log_a(x) \quad (a > 0, a \neq 1)$

• Domain: $(0, \infty)$
• Range: $\mathbb{R}$
• $x$-intercept: $(1, 0)$
• Vertical asymptote: $x = 0$
• Inverse of exponential function

Function Operations

Arithmetic Operations

$(f + g)(x) = f(x) + g(x)$

$(f - g)(x) = f(x) - g(x)$

$(f \cdot g)(x) = f(x) \cdot g(x)$

$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, \quad g(x) \neq 0$

Composition

$(f \circ g)(x) = f(g(x))$

$(g \circ f)(x) = g(f(x))$

Note: Composition is not commutative: $f \circ g \neq g \circ f$ (in general)

Inverse Functions

A function $f$ has an inverse $f^{-1}$ if and only if $f$ is one-to-one (passes horizontal line test).

$f(f^{-1}(x)) = x$

$f^{-1}(f(x)) = x$

Finding the Inverse

1. Replace $f(x)$ with $y$
2. Swap $x$ and $y$
3. Solve for $y$
4. Replace $y$ with $f^{-1}(x)$

Function Properties

Property	Definition
Even	$f(-x) = f(x)$
Odd	$f(-x) = -f(x)$
One-to-one	$f(a) = f(b)$ implies $a = b$
Onto	Every element in codomain is mapped
Increasing	$x_1 < x_2$ implies $f(x_1) < f(x_2)$
Decreasing	$x_1 < x_2$ implies $f(x_1) > f(x_2)$