A function is a rule that assigns each input exactly one output. It is written as f(x) = expression, where x is the input (domain) and f(x) is the output (range). The key test: each x-value must map to only one y-value — this is the vertical line test on a graph.
Linear functions have the form f(x) = mx + b and produce straight-line graphs. Quadratic functions have the form f(x) = ax² + bx + c and produce parabolas. Both are types of polynomial functions.
Function composition combines two functions: (f ∘ g)(x) = f(g(x)). You first apply g to x, then apply f to the result. This creates a chain of operations.
Inverse functions reverse the effect of the original function. If f(x) = 2x + 3, then f⁻¹(x) = (x - 3) / 2. Graphically, the inverse is a reflection across the line y = x.