Definition of a function

What is a function in math? Basic function definition

A function is a relation that maps a set of inputs to a set of outputs and each input is related to only one output.

The main word in the definition of a function is the "relation". If we have determined this relation, then we have determined the function.

Notice, two inputs can be related to the same output.

But each input is related to just one output.

And opposite, one input can't be related to more then one output.


There is a function

y = f(x)


  • x (the input to the function) is called an independent variable or argument of the function,
  • y (the output of the function) is called a dependent variable or function,
  • f is a name of this function.

We read it "y equals f of x" or "y is a function of x".

See Function notation.

Example of a function.

f(x) = 3x


  • x is the independent variable or argument,
  • f is the function name,
  • the relationship is the "multiply by 3".

This function can be written so

y = 3x


  • y is the dependent variable or function,
  • the function has no name in this case.