# Definition:Real Function/Definition by Equation

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## Definition

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 see

- Results about
**Real Functions**can be found here.

## Sources

- 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term*Function of One Independent Variable*: Comment $2.33$ - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.3$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 20$: Introduction: Remarks $\text{(i)}$