Inverse function

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chaarleey

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Hi!
Need help with a problem:

Prove that y=x^7+5x^3+3 is invertible and find the derivative of f^-1 in the point 9
 
chaarleey said:
Hi!
Need help with a problem:

Prove that y=x^7+5x^3+3 is invertible and find the derivative of f^-1 in the point 9
Click to expand...
Hi, The rules of the forum is that you receive help only after showing your attempt. So please show us your work, even if you know it is wrong.
What does it mean for a function to be invertible? How do you go about showing that a function is invertible? What is the formula to find the derivative of f^-1?
 
All i know really about inverse is to exchange x for y and then solve for y. But i get stuck on trying that. i guess there is another way of doing this?
 
chaarleey said:
All i know really about inverse is to exchange x for y and then solve for y. But i get stuck on trying that. i guess there is another way of doing this?
Click to expand...

Hi
Possibly reading
http://en.wikipedia.org/wiki/Inverse_function
would help. One of the things implied there it that you do not have to be able to explicitly define the inverse function for it to exist. In fact, sometimes that explicit inverse [exchanging x for y and then solve for y] can not be accomplished.

There are several ways to go about determining whether an inverse exists and to exchange x for y and then solve for y is only one of them. In this case, possibly the easiest way to prove that the f(x),
f(x) = y = x7 + 5 x3 + 3,
has an inverse is to use the fact that a (continuous) function (whose derivative exists) has an inverse if and only if it has no local extrema [it's derivative is either non-negative in its domain or it is non-positive in its domain]. If we let g(x) [= f-1(x)] denote that inverse and if f'(x) is not zero,
g'(x) = 1/f'(x)
 
So if f´(x)= 15x^2+7x^6 , it is positive which means it has a minimum but no maximum?
 
chaarleey said:
So if f´(x)= 15x^2+7x^6 , it is positive which means it has a minimum but no maximum?
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As you indicate
f'(x) = 15 x2 + 7 x6
So what is f'(0)? Is it a relative maximum or relative minimum or just a point of inflection?
 
f´(0) would just be 0. by looking at the graph i can see that it is the minimum but how do i go about proving that? would that be enough to say it has an inverse?
 
chaarleey said:
f´(0) would just be 0. by looking at the graph i can see that it is the minimum but how do i go about proving that? would that be enough to say it has an inverse?
Click to expand...
Yes, x=0 would be a relative (and global) minimum for f'. However, I may have been unclear because what you need to look at is the function f itself. Is x=0 a relative maximum or relative minimum or just a point of inflection for the function f? How would you go about proving it? [and don't forget the second part of the question]
 
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