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A fuction is a relation in which no two of the ordered pairs have the same first element. Every set of ordered pairs is a relation but every relation is not a function. Functions make up a subset of all relations.
To determine if a relation is a function, use the vertical line test
1) Sketch the graph of the relation.
2) Make a visual check of the numbers of times a vertical line would cut the graph.
3) If the vertical line only ever cuts at one place, the relation is a function. If the vertical line cuts at two or more places ( for the same x-value), then the relation is not a function.
Mapping and Function Notation
A relation is also called a mapping if, for example, every element is a set X is related to exactly one element of a second set Y.
Such mappings are often called functional relations, or simply functions. So that is fact, a function is really another name for a mapping.
A function f, (of a mapping f), from a set X to a set Y is a relation which assigns to each element x of the set X a unique element y of the set Y, the co-domain. The set X is called the image of x under f and we denote this image by f(x) (read as; f of x) the value of the function f at x. We write f: x --> f(x).
We can also write these expressions as follows
f: X |--> Y, where y = f(x)
f: X |-->Y, y = f(x)
y = f(x), x E X
The range of f is not necessarily the set Y. The range of f is actually a subset of Y (or it could be equal to Y).Set Y describes the types of numbers that will be produced when f is applied to different x- values – not necessarily which numbers we will end up with. So the range of f is given by the values of f(x).
