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 from A to B, where A is the domain containing all inputs, and B is the codomain containing all possible outputs. The range is the set of actual outputs used.
Function notation uses f of x equals y to represent the relationship between input and output. Here, 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 the input value and calculate the result step by step.
Functions can be represented visually through coordinate graphs, mapping diagrams, and tables. The vertical line test helps determine if a graph represents a function. If any vertical line intersects the graph at most once, it's a function. A parabola passes this test, but a circle fails because vertical lines can intersect it twice.
There are several major types of functions. Linear functions have the form f of x equals mx plus b and create straight lines. Quadratic functions follow f of x equals ax squared plus bx plus c, forming parabolas. Exponential functions use f of x equals a to the x power, showing rapid growth or decay. Each type has distinct characteristics and graph shapes that help identify them.
Functions model many real-world relationships. Temperature conversion uses C equals five-ninths times F minus 32. Compound interest follows A equals P times one plus r to the power t. Distance equals rate times time. These functions help solve practical problems by showing how changing inputs affect outputs in predictable ways.