Definition:Real Function/Definition by Formula
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 $x \in S$.
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.
- Results about Real Functions can be found here.