Question

Define a binary relation P from R to R as follows: for all real numbers x and y, (x,y)∈P⇔x=y^2. Is P a function? Explain.

Answer 1

Taking into account that $4=2^2$ and $4=(-2)^2$, we conclude that $(4,2)\in P$ and $(4,-2)\in P$. Therefore, the element $x=4\in \mathbb R$ corresponds to two elements $y=2\in\mathbb R$ and $y=-2\in\mathbb R$, and hence $P:\mathbb R\to\mathbb R$ is not a function $y=f(x).$

On the other hand, for each element $y\in\mathbb R$ there exists a unique element $x=y^2\in\mathbb R$. Consequently, $P:\mathbb R\to \mathbb R, P(y)=y^2,$ is a function of variable $y.$