Functions and Mappings

Domain and Range

• Domain: set of allowed inputs.
• Range: set of possible outputs.

$f(x) = \sqrt{x}$: domain $x \geq 0$, range $f(x) \geq 0$.

Types of Mapping

• One-to-one: each output from at most one input. Has an inverse.
• Many-to-one: some outputs from multiple inputs. No inverse unless domain restricted.

Composite Functions

$fg(x) = f(g(x))$. Domain of $fg$: values in domain of $g$ where $g(x)$ is in domain of $f$.

Inverse Functions

Only one-to-one functions have inverses. The graph of $f^{-1}$ is the reflection of $f$ in $y = x$.

Finding: swap $x$ and $y$, rearrange.

$f(x) = \frac{2x+1}{x-3}$. Swap: $x = \frac{2y+1}{y-3}$ → $xy - 3x = 2y + 1$ → $y(x-2) = 3x+1$ → $f^{-1}(x) = \frac{3x+1}{x-2}$.