Question

Suppose $f$ is the collection of all ordered pairs of real numbers and x = 6 is the first element of some ordered pair in $f$. Suppose the vertical line through x = 6 intersects the graph of $f$ twice. Is $f$ a function? Why or why not?

Answer 1

f is not a function.

a function f is the same thing as a subset $G_f$ of $X\times Y$ with the following property:

for all $x\in X$, there is a unique $y\in Y$ such that $(x,y)\in G_f$ and we set y = f(x).

To say that there is a unique $y\in Y$ says that f(x) is uniquely determined by x, and to say that for every $x\in X$ there exists an $(x,y)\in G_f$ says that in fact f(x) is defined for all $x\in X$.

This is the so-called vertical line test: for each $x\in X$, we have the subset $x\times YofX\times Y$. (In case $X=Y=R$, such subsets are exactly the vertical lines.)

Then G is the graph of a function f if and only if, for every $x\in X$, ({x} × Y ) ∩ G consists of exactly one point, necessarily of the form $(x,y)$ for some $y\in Y$ .

The unique such y is then $f\left(x\right).$