A function is a fundamental concept in mathematics. It's a special type of relation that connects inputs to outputs. The key rule is that each input must correspond to exactly one output. Think of it like a machine: you put something in, and you get exactly one thing out.
Every function has a domain and range. The domain is the set of all possible input values that the function can accept. The range is the set of all possible output values. For example, the function f(x) = x² has a domain of all real numbers because we can square any real number. However, its range is only non-negative numbers because squaring always gives a positive result or zero.
The most important rule that defines a function is the one-to-one correspondence between inputs and outputs. Each input value must produce exactly one output value. This is what distinguishes a function from other types of relations. If you have an input that leads to multiple outputs, then you don't have a function. This rule ensures that functions are predictable and well-defined.
Function notation is a standard way to write and work with functions. We use f(x) to represent a function, where f is the name of the function and x is the input variable. The expression f(x) represents the output value. For example, if f(x) equals 2x plus 1, then f(3) means we substitute 3 for x, giving us 2 times 3 plus 1, which equals 7. This notation makes it clear what the input is and what the corresponding output will be.
To summarize, a function is a mathematical relationship that assigns exactly one output to each input. The key characteristics are: the domain which contains all possible inputs, the range which contains all possible outputs, and the one-to-one rule that ensures each input produces exactly one output. Functions can be written using function notation like f(x), and they come in many types including linear, quadratic, and trigonometric functions. Understanding functions is essential because they are fundamental tools used throughout mathematics, science, and engineering to model relationships between quantities.