Help with differentiatable function question!

T

TheShadow

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Find all differentiable functions \(\displaystyle \,f\, :\, \mathbb{R}\, \rightarrow\, \mathbb{R},\) such that \(\displaystyle \,f\, \circ\, f\, =\, f.\)
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I came across this question and I've been struggling badly trying to solve. :( can someone please help me out, with some working out? I'd really appreciate it!

Thanks!
 
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TheShadow said:
I came across this question and I've been struggling badly trying to solve. :( can someone please help me out, with some working out? I'd really appreciate it!
Click to expand...
What have you tried so far? For instance, you took the derivative of both sides of the equality, and considered cases. If f'(x) = 0, then what just f(x) be? If not, then what? And so forth.

Please be complete. Thank you! ;)
 
stapel said:
What have you tried so far? For instance, you took the derivative of both sides of the equality, and considered cases. If f'(x) = 0, then what just f(x) be? If not, then what? And so forth.

Please be complete. Thank you! ;)
Click to expand...

im actually really lost. I mean I don't know how to go about doing this, it's very confusing. Can you explain further what you're talking about? ^
 
TheShadow said:
im actually really lost. I mean I don't know how to go about doing this, it's very confusing. Can you explain further what you're talking about? ^
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Why not try using what you've already been given? That is:

What did you get when you differentiated both sides?
 
TheShadow said:
I managed to get f'^2 (x) = f' (x)

Is that right?
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No. What does the Chain Rule say? Does it say that the derivative of h(g(x)) is (h'(x))^2 or (g'(x))^2? Or does it say that the derivative is h'(g(x)) * g'(x)?

Apply the Chain Rule. Then see what you can solve for, and under what conditions. ;)
 
TheShadow said:
I came across this question and I've been struggling badly trying to solve. :( can someone please help me out, with some working out? I'd really appreciate it!

Thanks!
Click to expand...
Out of curiosity, is that f times f equal f, i.e. f2 = f, or the composite of f with f equals f, i.e. f(f(x)) = f(x)?



In the one case (composite) you have
>>>>>>>>>>>>>>>
f' f' = f'
and either
f' is zero or
f' = 1
If f' is zero then f is a constant and
f(x) = c
f(f(x)) = c.

If f' is not zero, then
f(x) = x + c
where c is a constant. So
f(f(x)) = (x + c) + c = x + 2 c = f(x) = x + c
and c is zero so
f(x) = x.
<<<<<<<<<<<<<<<

Edit: Whoops - looks like it was the composite and I shouldn't do the problem unless the OP is really stuck.
 
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