A function is a fundamental concept in mathematics. It's a special type of relation between two sets where each input has exactly one output. We write this as f colon A arrow B, where A is the domain containing all possible inputs, and B is the codomain containing all possible outputs. The key property is that every element in set A maps to exactly one element in set B.
Function notation uses f of x equals y, where x is the independent variable or input, and y is the dependent variable or output. Common examples include f of x equals 2x plus 1, g of t equals t squared, and h of n equals 1 over n. To evaluate a function, we substitute specific values. For instance, if f of x equals 2x plus 1, then f of 3 equals 7, f of 0 equals 1, and f of negative 2 equals negative 3.
Functions can be visualized in multiple ways. Coordinate graphs show the relationship between input and output values. The vertical line test helps determine if a graph represents a function - if any vertical line intersects the graph more than once, it's not a function. For example, y equals x squared is a function, but x squared plus y squared equals 4, which is a circle, is not a function because vertical lines can intersect it twice.
There are several important types of functions. Linear functions like f of x equals mx plus b create straight lines. Quadratic functions such as f of x equals ax squared plus bx plus c form parabolas. Exponential functions like f of x equals a to the x show rapid growth or decay. Trigonometric functions including sine and cosine create periodic wave patterns. Each type has distinct characteristics that help us identify and work with them.