Relationship of Y^2=f(x) and Y=f(x)

[mune]

hi guys, what is the relationship between y^2 = f(x) and y = f(x)?

thank you.

 

[symbolipoint]

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.

 

[d_leet]

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.

Or y=0...
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[mune]

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.

 

[mune]

it is not a trick question, nor a very highly theoretical question.

 

[d_leet]

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.

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 let's let
y=f(x) and
z=(f(x))²

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

y'=f'(x) and
z'=2f(x)f'(x)

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.

 

[HallsofIvy]

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?

 

[Gib Z]

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.

 

[mune]

thank you everyone
sorry that I didn't explain my question clearly. Anyway, Gib Z and HallsofIvy know what I mean

but thanks d_leet too, I have learned a way to prove from your post.

 

[daveb]

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.

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).