2.5 Vertical Line Test

Not every curve in the plane is the graph of a function. For example, consider the curve shown in Figure 1. In this figure, a vertical line intersects the curve at two distinct points $(a,b)$ and $(a,c)$. This curve does not represent any function. If this curve were the graph of a function $y=f(x)$, it would mean that
$$b=f(a),c=f(a),$$ which is impossible as the function $f$ cannot output two different values for the same input $a$. So we have a simple test, called the vertical line test, that says if any arbitrary vertical line intersects a curve more than once, that curve cannot be the graph of a function.