A relation is a function if each input x is associated with exactly one output y. For example, the equation
y = 2x + 5 defines a function because every value of x yields a unique y. However, x = y² + 1 is not a function of x, since a single x-value, such as x = 2, corresponds to two possible y-values: y = 1 and y = -1.
The vertical line test helps determine whether a graph represents a function. If a vertical line intersects a curve more than once, the curve fails the test and does not represent a function. For instance, the graph of x = y³ − y is not a function of x, as it loops back on itself. On the other hand, 
is a function, since for every x ≥ − 4, there is only one value of y.
Some non-functions can be split into two valid functions. For example, the graph of y² = x is not a function, but splitting it into 
and 
produces two functions.
Finally, in a table of values, if a single input appears more than once with different outputs, the table does not represent a function.
An equation defines a function when each input from the domain gives only one output in the range. If a single input from the domain gives two different outputs from the range, the relation is not a function.
Function tables follow the same rule, if a table gives more than one output for the same input, it does not define a function.
On a graph, if two points share the same x-value but have different y-values, then it is not a function.
A graph represents a function if it satisfies the vertical line test, meaning no vertical line intersects the graph more than once.
For example, a sideways parabola fails the vertical line test because vertical lines intersect it at two points.
Splitting this parabola into upper and lower parts creates two separate functions. Each one passes the vertical line test.
Reversing the roles of x and y in a sideways parabola can define x as a function of y as long as each y-value corresponds to only one x-value.
For example, a working vending machine is a great illustration of a function: each button pressed from the given domain corresponds to one and only one specific snack from the range.
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