"We are given the relation:" \qquad \qquad x \ =\ y^2.
"We are asked to decide if it defines a function."
"If no matter what the value of the first variable," \ x, "there is"
"precisely one value of the second variable," \ y, "connected"
"to it inside the relationship -- then it will be a function. If this"
"breaks down for even one value of the first variable, it will fail"
"to be a function. That is to say, if for some value of the first"
"variable, there are two or more values (or no values) of the"
"second variable connected to it inside the relationship, then it"
"will not be a function."
"Note -- in general, there is no procedure to decide if an"
"arbitrarily given relation is functional [ -- is a function or not]."
"The truth is, in general, there are no such procedures. Our"
"case, thankfully, turns out to be simple enough to make the"
"decision, let's say, using good instincts!! "
"We have:" \qquad \qquad x \ =\ y^2.
"We ask, in our mind, for a given value of" \ \ x, "how many values"
"of" \ \ y \ \ "are connected to it in the relationship -- one, or more"
"than one ?"
"That is to say, for a given value of" \ x, "how many solutions" \ \ y \ \
"are there to the relation:" \ x \ = \ y^2 \ \ "? -- one, or more than one ?"
"For example, for" \ \ x \ \ "taking the value" \ 1, "how many solutions" \ \ y
"are there to the resulting relation:" \qquad \qquad \underbrace{1}_{x} \ = \ y^2 \ \ "?"
" -- one, or more than one -- "?"
"This is, thankfully (!), easy to decide !! We proceed, looking"
"at the solutions of:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 1 \ = \ y^2.
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad y^2 \ = \ 1.
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad y \ = \ \pm sqrt{1}.
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad y \ = \ -1, 1.
"So, for" \ \ x \ \ "taking the value" \ 1, "there are two values for" \ \ y \ \
"connected to it in the given relation:" \ -1, 1 . \ \ "So, more than"
"one value for" \ \ y, \ "for this value of" \ \ x. \ \ "This ends the decision"
"right here."
"We can stop immediately now -- and conclude that the given"
"relation is not a function."
"This is our result:"
\qquad \qquad \qquad \qquad \quad "the relation" \qquad x \ = \ y^2 \qquad "is not a function."
"I want to make a perhaps valuable note, to keep perspective."
"If in the above work, we had picked the value of" \ \ 0 \ \ "for" \ \ x \ \
"to take in the relation, and then looked to see how many"
"solutions" \ \ y \ \ "there are to the resulting relation:" \ \ 0 \ = \ y^2,
"we would have looked at the solutions of:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 0 \ = \ y^2.
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad y^2 \ = \ 0.
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad y \ = \ 0, \quad "only".
"And we would have concluded that, for" \ \ x \ \ "taking the value" \ 0,
"there is exactly one value" \ \ y \ \ "connected to it in the given"
"relation:" \ \ 0. \ \ "Exactly one value for" \ \ y, \ "connected to this"
"value of" \ \ x.
"What does this tell us about whether the given relation is a"
"function ? NOTHING !!"
"Because there is exactly one value for" \ \ y \ \ "for this value of" \ \ x,
"we cannot exclude the relation from being a function, as we did"
"above using the value of" \ \ 1 \ \ "for" \ \ x.
"We also cannot say from this that the relation is a function,"
"either. Why ? The work here told us what happened with the"
"values for" \ \ y \ \ "connected with the value" \ \ 0 \ \ "for" \ \ x \ \ "-- exactly one"
"value for" \ \ y. \ \ "But it told us nothing about the values for" \ \ y \ \ "
"connected with any other value for" \ \ x. \ \ "Other values for"
\ \ x \ \ "might have exactly one value for" \ \ y \ \ "connected to it, "
"might have more than one value for" \ \ y \ \ "connected to it, or"
"might have no values for" \ \ y \ \ "connected to it. We cannot know"
"unless we go back and check values for" \ \ x, "other than" \ \ 0."
"What other values for" \ \ x, "should we check -- other than" \ \ 0 \ \ "?"
"The truth is, in general, there is no way to determine what"
"other values for" \ \ x \ \ "(if there are any) we should check. We"
"were lucky we picked the value" \ \ 1 \ \ "for" \ \ x \ \ "above -- which"
"allowed us to make a decision on this relation. For certain"
"types of relations, there are ways to determine other values"
"to check. In general, there is no such procedure for finding"
"such luck -- just hope, and good instincts !!"
