Function
Main Concept

Recall that a binary relation on sets of objects $A$ and $B$ is a collection of ordered pairs $\left(a\,b\right)$, where $a$ is one of the objects in $A$ and $b$ is one of the objects in $B$. A function is a binary relation with the additional property that each object in $A$ occurs as the first component of at most one pair in the relation.
The domain of a function is the set of all values which appear as the first component in the function. The range of a function is the set of all values which appear as the second component in the function.
A note on notation
Functions are fundamental objects in mathematics, and there is a variety of notations in common use for representing and describing functions. For example, the squaring function given by the ordered pairs $\left(x\,{x}^{2}\right)$, where x is any number, could also be written as:
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$f\left(x\right)\={x}^{2}$

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$f\:x\to {x}^{2}$

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$x\stackrel{f}{\to}{x}^{2}$

The last notation shown is very common, and arises from the standard convention of labeling the vertical axis of a graph with the letter y (and the horizontal axis with x).



The requirement that each input to a function must produce exactly one output can be used to visually determine when a graph is the graph of a function. Points on the horizontal axis represent the inputs and points on the graph represent input/output pairs. So, if the graph represents a function then a vertical line through any point on the horizontal axis can intersect the graph at most once. This visual check is known as the vertical line test.
Click and drag in the plot below to draw a graph. A vertical line drawn with your graph will show whether your graph represents a function. Clear an existing graph by clicking it.

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