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 $x \in S$.
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 $x$.
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.
- 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text I$: Differentiation: Functions
- 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 $2.32$
- 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