Prove Graph is Function: Logic & Alternatives

[Mentallic]

I'm interested in getting a better answer to showing/proving a graph is a function than the usual "horizontal line test" response. Is the use of logic the best tool for proving a function? Because that line test method is basically that.
Also, translating the function by illustrating it on the x-y plane isn't always so simple without a calculator, so other methods may be necessary.

 

[Office_Shredder]

Depends on what you want to do. Are you trying to show a graph is representative of a function (in which case you don't need to worry about trying to draw it) or are you trying to show that an equation of the form f(x,y) = 0 i.e. an equation with x's and y's can be written so that y is a function of x?

 

[Mentallic]

I'm trying to show the second that you mentioned

or are you trying to show that an equation of the form f(x,y) = 0 i.e. an equation with x's and y's can be written so that y is a function of x?

 

[Office_Shredder]

What you really need to show is:

Given a value of x, there is only one possible value of y(x). Example for each:

x² + y² = 1

If x = $\frac{1}{\sqrt{2}}$ then y can be plus or minus $\frac{1}{\sqrt{2}}$ so this does not give a function (and drawing the circle we see this is obvious). More generally, if y² = 1-x² and w² = 1-x² for the same value of x, then y² = w² so y= +/- w and in general y and w do not have to be equal

If on the other hand we have 2y-x-1 = 0 Then if 2y-x-1 = 0 and 2w-x-1 = 0 for a set value of x, then 2y=x+1=2w and we can see y=w is necessary. Hence for each value of x there is only a single value of y that satisfies this, and y can be defined as a function of x.

There's also another possible concern about the domain. y is a function of x over the real numbers by definition also means for each x, there exists a possible value of y. If your possible function was $y = \sqrt{1-x^2}$ (taking only positive values of square root) unlike in the circle case this satisfies the condition for each x there is at most one value of y. But by definition, this is not a function over the real numbers because outside of the interval [-1,1] y is not defined. Instead it's only a function over that interval. Generally this isn't the condition you're worried about though (since what the domain of the function is probably isn't the question at hand)

 

[Mentallic]

So it is necessary to make the y variable of an equation f(x,y) the subject so as to know if it is the form $$y^2=y(x)$$ or maybe other operative functions that result in more than one value such as $$sin(y)=y(x)$$ or others that exist (which I wouldn't know about).
Is it possible to show the equation is a function or not without re-arranging the equation? Possibly because re-arrangement would be too difficult or impossible with elementary functions.