how to define a function?

Question

If we define a function by set theory it states that it is relation in the sets of inputs and outputs such that each input is exactly related to one out put . so if $A=\{9,25,36\} ;\, B=\{3,5,6\}$ and a relation $R=\{(x,y)|\pm\sqrt{x}=y\,\land \,x\in A,\,y\in B\}$ also a function because it has exactly one element in $B$ for $A$ . But if we define a function without set theory it states that a function produces one output for every input then the above mentioned relation is not a function . so which is right and why ? please explain it .

Answer 1

See more: https://en.wikipedia.org/wiki/Function_(mathematics)

A function is defined by an input-output relation; there must exist an ordered pair for $(x,y) \in X \times Y$.

Moreover, your relation is not a function because of the $+/-$ sign in front of $\sqrt{x}$; assuming one substitutes a certain value of $x$ one can get two different values for $y$.