Definition:Real Function/Definition by Formula

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A function $f: S \to T$ can be considered as a formula which tells us how to determine what the value of $y \in T$ is when we have selected a value for $x \in S$.

As an Equation

It is often convenient to refer to an equation or formula as though it were a function.

What is meant is that the equation defines the function; that is, it specifies the rule by which we obtain the value of $y$ from a given $x$.

Square Function

For example, let $x, y \in \R$.

The (real) square function is the real function $f: \R \to \R$ defined as:

$\forall x \in \R: \map f x = x^2$

We may express this as $y = x^2$, and use this equation to define this function.

This may be conceived as:

For each $x \in \R$, the number $y \in \R$ assigned to it is that which we get by squaring $x$.

Another useful notation is:

$\forall x \in \R: x \mapsto x^2$

Also known as

Some sources, possibly in an attempt to improve the accessibility of the subject, refer to the formula for a function as a recipe.

Other sources use the term rule.

Also see

  • Results about Real Functions can be found here.