1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Y^2=f(x), y=f(x)

  1. Aug 7, 2007 #1
    hi guys, what is the relationship between y^2 = f(x) and y = f(x)?

    thank you.
  2. jcsd
  3. Aug 8, 2007 #2


    User Avatar
    Homework Helper
    Education Advisor
    Gold Member

    Is that a trick question or is it a very highly theoretical and advanced question? You have indicated a situation in which y^2 = y. Best conclusion is y=1 and f(x)=1, horizontal line, one unit above the x axis.
  4. Aug 8, 2007 #3
    Or y=0...
    /*extra characters*/
  5. Aug 8, 2007 #4
    let's say we are given a function y=f(x), what is relationship between it and y^2 =f(x)?

    for example, if y=f(x) have a maximum point at (a,b), will y^2=f(x) have a maximum point at(a,b) too?

    i hope i make my question clear.:smile:
  6. Aug 8, 2007 #5
    it is not a trick question, nor a very highly theoretical question.
  7. Aug 8, 2007 #6
    Oh, you don't really mean y=f(x), and y2=f(x) do you? I think what you mean to say is if we're given a function f(x), then what is the relation between the function f(x), and the function (f(x))2.

    so lets let
    y=f(x) and

    Obviously z is always positive assuming we are only dealing with real numbers, but to investigate a relation about maxima/minima lets look at y'.

    y'=f'(x) and

    if f(x) has a max/min at the point (a,b) then f'(a)=0
    Then y'(a)=0, and I think you can see then that z'(a)=2f(a)f'(a), but f'(a)=0 so z'(a)=0, thus the function z or (f(x))2 will also have a max or min at the point a, however this time it will be at the point (a, b2).

    I think it should be fairly easy to show that if f has a min at x=a then so does f2, and the same if f has a max at x=a, but I'm a bit too tired to try a proof of that at the moment.
  8. Aug 8, 2007 #7


    User Avatar
    Science Advisor

    No, that's not how I would interpret the question.

    Let's clarify by taking f(x)= x. What is the relationship between y= x and y2= x?

    Suppose f(x)= x+ 3. What is the relationship between y= x+ 3 and y2= x+ 3?

    Frankly, I don't see much relationship. The first is a function and the second, for general f(x), is NOT a function. The first might be a square root of the second (if the first is positive for all x) but that was obvious wasn't it?
  9. Aug 8, 2007 #8

    Gib Z

    User Avatar
    Homework Helper

    I am quite sure this question is derived from a common one I've seen: Given a sketch of the graph of y=f(x), sketch y^2=f(x) labeling important features. d_leet's post works on that a bit. Also remember to find all points where y= 0 or 1, the graphs intersect there. Between zero and one, the y^2 graph will be slightly above the y graph. Other values, it will be below. You know the y^2 graph is discontinuous at the points where the y graph is negative.
  10. Aug 8, 2007 #9
    thank you everyone :smile:
    sorry that I didn't explain my question clearly. Anyway, Gib Z and HallsofIvy know what I mean :cool:

    but thanks d_leet too, I have learnt a way to prove from your post.
    Last edited: Aug 8, 2007
  11. Aug 8, 2007 #10
    That depends on whether f(a) is positive or negative. If f(a) is negative and a minimum, (f(a))^2 may be a maximum (e.g., if f(x) is the cosine function, then f(pi) is a minimum put (f(pi))^2 is a maximum).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook