# "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 !!" #
