# Definition:Real Function/Definition by Formula

## Definition

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.

## Sources

- 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.2$: Functions - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.10$: Functions