A function in mathematics is a rule or expression that defines a relationship between two variables, where each input (independent variable) corresponds to exactly one output (dependent variable). In other words, a function assigns to each element in its domain exactly one element in its codomain or range
. Key points about functions include:
- The independent variable is the input value, often represented as xxx.
- The dependent variable is the output value, often represented as f(x)f(x)f(x) or yyy, which depends on the input.
- For every input, there is only one output; no input can produce multiple outputs.
- The domain is the set of all possible inputs for the function.
- The range (or image) is the set of all possible outputs produced by the function.
An example of a function is f(x)=x+3f(x)=x+3f(x)=x+3, which adds 3 to any input number xxx, producing a unique output
. Graphically, a function can be identified because any vertical line drawn on its graph intersects the graph at most once, reflecting the one-output-per- input rule
. Thus, a function can be thought of as a "machine" that takes an input, applies a rule, and produces a single output
