Thread: How to prove x^2 is a function?

Title says it all. How do you prove x^2 is a function *besides* the vertical line test?

Originally Posted by vandrop

Title says it all. How do you prove x^2 is a function *besides* the vertical line test?

You need to know what the definition of a function is. Then, you need to consider what short-cut the vertical line test provides in applying the definition.

Originally Posted by mr fantastic

You need to know what the definition of a function is. Then, you need to consider what short-cut the vertical line test provides in applying the definition.

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 and , with . 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.

How to prove x^2 is a function?

Also can you answer to questions such as :
- How to prove 3x+7 is a function?
- How to prove x+7 is a function?
- How to prove 7 is a function?
- How to prove 0 is a function?

Originally Posted by JJacquelin

Also can you answer to questions such as :
- How to prove 3x+7 is a function?
- How to prove x+7 is a function?
- How to prove 7 is a function?
- How to prove 0 is a function?

Obviously either the vertical line test or the method outlined by HallsofIvy in post #7! I'd strongly advise the former given the level you appear to be studying at.

Originally Posted by HallsofIvy

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 and , with . What can you say about x?

i'd like to point out that there is no difference between a function and its graph. that is, formally, a function is a certain kind of relationship between two sets,

so the set of all points {(x,f(x)} which define f is what we mean (and draw) by the graph of f.

Originally Posted by mr fantastic

Obviously either the vertical line test or the method outlined by HallsofIvy in post #7! I'd strongly advise the former given the level you appear to be studying at.

Ha! ha! Funny !
But you didn't answer to the question "How to prove 0 is a function?"
Of course, I know how to answer to this question. But You ?

Originally Posted by JJacquelin

Ha! ha! Funny !
But you didn't answer to the question "How to prove 0 is a function?"
Of course, I know how to answer to this question. But You ?

If you know the answer to the questions you post, you're wasting the time of people who might reply in good faith, thinking they are helping someone.

If you want to challenge members with a question (regardless of how trivial the question is), post the question here: http://www.mathhelpforum.com/math-help/f25/

after reading the rules for posting in that subforum.