Title says it all. How do you prove x^2 is a function *besides* the vertical line test?
Indeed... x^2 is the example used in the wikipedia page on functions !
the formal definiton of a function is: f(x) ≠ f(y) --> x ≠ y (well more specifically, (x1,x2) in f, and (x1,x3) in f -->x2 = x3, but these two statements are equivalent).
so suppose that x^2 ≠ y^2. this means that x^2 - y^2 ≠ 0. hence (x + y)(x - y) ≠ 0.
in particular NEITHER x + y nor x - y can be 0. so x - y ≠ 0, and thus x ≠ y.
in the other form, suppose (x,x^2) and (x,y^2) are in f = {(a,b) in R x R : b = a^2}. then y^2 = x^2.
it is sometimes helpful to see why some relations fail to be functions. suppose g = {(a,b) in R x R : a = 1}.
(this is the vertical line x = 1). well, we have (1,2) in g, and (1,3) in g, but clearly 2 ≠ 3.
therefore, we have no idea what value to assign to g(1), we have "too many choices".
this is why we write f(a) for the value of a function f at the point a, there is only one possible number we could pick, if f is a function.
A function essentially is saying that whenever you input a number for x you only get one answer for y. Anything else would be illogical.
I view it like a coffee machine. When I put the coffee grinds and water in, I expect to get coffee every time. Not coffee the first time, orange juice the second and hot chocolate the third time. If that starts happening my machine is not functioning properly.
Same thing with function equations:
y = x^2
If I am inputting 2 into this function then I should get a y=4 EVERY time. Otherwise something is wrong. It is logical to see this equation and realize that you cannot plug in a number for x and get two different answers. Not a formal proof, but a little easier to understand why it works worded this way I think.
The non example Deveno was talking about referred to a vertical (up and down) line.
If you take a look at the points (ordered pair) of a vertical line all of the x-coordinates (inputs) are the same and the outputs are different. You might see ordered pair like this:
(1,2)
(1,3)
(1,4)
etc
So a vertical line is not a function because every time you put in 1 for x you get a different answer for y. This is illogical.
A simple way to "prove" that y = x^2 is a function is that it moves from left to right forever and ever and does not stop its forward motion to go straight up or turn around. These qualities mean prove that x will never get you two different answers for y because you are always moving left or right so your x-coordinates will never repeat.
That makes no sense to me at all. Functions do not "move" at all. I suspect you are talking about the graph of f(x) but it would be a good idea to say that! In any case, you are really using the "vertical line" test and vandrop said "*besides* the vertical line test".
Suppose $\displaystyle x^2= y_1$ and $\displaystyle x^2= y_2$, with $\displaystyle y_1\ne y_2$. What can you say about x?
To me the vertical line test is physically taking a vertical line and passing it along the graph and looking for it to cross two points of the line at the same time. Which is a different mindset than just looking to see if the line, curve or wave is always moving left to right without stopping/turning around and not using the vertical line to accomplish this.
They are both essentially accomplishing the same thing so I guess I can see how they are thought of as the same thing.
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