# Is  x=3 a function?

Jul 17, 2015

No. The equation $x = 3$ represents a vertical line and not a mapping from values of $x$ to values of $y$.

For one thing, it fails the vertical line test for functions at the value $x = 3$.

#### Explanation:

A function maps $x$ values to $y$ values, in such a way that there is exactly one $y$ value for each $x$ value.

For a given $y$ value, there may be several $x$ values that map to it, but for any given $x$ value, there cannot be more than one corresponding value of $y$.

One way to express this condition is the vertical line test. Any vertical line must intersect the curve described by the equation at at most one point. If it does not intersect the curve at all, then the corresponding $x$ value is not part of the domain of the function.

In the case of the curve (vertical line) described by $x = 3$, the only vertical line that intersects it at all is $x = 3$, and that intersects it at infinitely many points.
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