Language of Functions

Functions are typically written as $f(x) = ...$ or using the mapping notation $f: x \mapsto ...$.

Understanding the set of numbers involved is crucial for defining a function's behavior.

Common sets include Natural Numbers ($\mathbb{N}$), Integers ($\mathbb{Z}$), Rational Numbers ($\mathbb{Q}$), and Real Numbers ($\mathbb{R}$).

Restricting a domain can change the nature of a mapping, often turning a many-to-one function into a one-to-one function.

This is mathematically significant because only one-to-one functions possess unique inverses.

For example, the rule $f(x) = x^2$ is many-to-one over $\mathbb{R}$, but becomes one-to-one if the domain is restricted to $x \geq 0$.

Check Validity: Always verify if a mapping is a function by checking if any input leads to multiple outputs; if it does, it fails the function test.

Sketching for Range: To find the range of a function, always sketch the graph within the given domain boundaries to identify maximum and minimum values.

Notation Precision: Use correct set notation (e.g., $f(x) \in \mathbb{R}, f(x) \geq 0$) when stating the range to ensure full marks.

Boundary Values: When a domain is restricted (e.g., $x > a$), check the value of the function at the boundary to determine if the range should use strict inequalities ($>$) or inclusive ones ($\geq$).