I am new to this forum. I look forward to conversing with you guys and gals. I do have a question
I am preparing for college course and I ran into this:
is is this a function and why or why not? C=Y^{2 }-2. Can someone explain
I am new to this forum. I look forward to conversing with you guys and gals. I do have a question
I am preparing for college course and I ran into this:
is is this a function and why or why not? C=Y^{2 }-2. Can someone explain
"C = Y^{2 } - 2'' is an equation, something that is either true or false when C and Y are given concrete numerical values. That's one thing that can be said for sure. It can be interpreted as a function, but this requires several choices, and not all of them lead to a function.
The first choice is to select the dependent and independent variables. If the independent variable is Y and the dependent one is C, then this is indeed a function mapping every Y into Y^{2} - 2. If the independent variable is C and the dependent one is Y, then this is not a function because some values of the independent variable correspond to two different values of the dependent variable. E.g., C = 2 corresponds to Y = 2 and Y = -2. But even in this case this can be turned into a function by artificially restricting the codomain, i.e., the set where the dependent variable ranges. If we postulate that Y is nonnegative, then the mapping from C to Y becomes a function again (only for C >= -2).
A function is not only a law that provides some output for each input, but it also includes the domain and the codomain, the sets where the input and output come from. The same law may produce different functions for different domains, and it can even be or not be a function depending the domain and the codomain, as we have seen above.
There are other ways to interpret this equality. We can treat it as a family of constant functions from X to Y, two for each C > -2. For example, C = 2 gives functions Y(X) = 2 and Y(X) = -2.
It seems to me that having a question about a "function" the first thing you would do is look up the definition of "function"! "A" is a function of "B" means that there is some rule, or other way of associating values of A and B such that a single value of B is associated with a unique value of A. Notice that is NOT "symmetric". We can have "A is a function of B" and NOT "B is a function of A".
So if I wanted to be really hard-nosed, I would say your question "is C= Y[sup[2[/sup] 2 a function" is meaningless because you haven't distinguished between "C" and "Y". If your question were "Is C a function of Y" then the answer would be "yes". Given any value for y, squaring it gives a single value and subtracting 2 from it again gives a single value (we say that the operations of "squaring" and "subtracting" are "well defined"). But if the question were "Is Y a function of C", the answer would be "no". For example if C= 2, then Y can be either 2 or -2.
Because our ways of writing "formulas" prefer "well defined" operations, whenever you have something that says "A= " with some operations involving, say, B, on the other side, you can be pretty sure A is a function of B- but B may not be a function of A.