Understanding Functions

A function is a special relationship that pairs each input with exactly one output. Functions can be represented in many ways: as equations, graphs, tables, or even words. The most common notation for a function is \( f(x) \), where \( x \) is the input and \( f(x) \) is the output.

Types of Functions

• Linear Functions: Straight-line graphs, like \( f(x) = 2x + 3 \).
• Quadratic Functions: Parabolic graphs, such as \( f(x) = x^2 - 4 \).
• Exponential Functions: Rapidly increasing or decreasing, like \( f(x) = 2^x \).
• Absolute Value Functions: Always non-negative, \( f(x) = |x| \).

Key Properties

• Domain: All possible inputs.
• Range: All possible outputs.
• Intercepts: Where the function crosses the axes.
• Symmetry: Even, odd, or neither.

Why Functions Matter

Functions help us model real-world scenarios, from tracking population growth to calculating profits.