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 $\text{(i)}$