1. ## inverse function problem

I have a function

$\displaystyle f(x) = 4x^2 - x^3$

If the domain of this function is restricted to 0 < x < 8/3 then f-1(x) can be defined - ie inverse function.

I am told to find the gradient of $\displaystyle f^-1(x)$ at the point ($\displaystyle \frac{7}{8}, \frac{1}{2}$)

My first thought was to work out the inverse function. But this looks hard? Is this what I should do? Or is there another way? I know that an inverse function is a reflection of function on y=x - can I somehow use this to calculate? If so how?

Angus

2. ## Re: inverse function problem

Originally Posted by angypangy
I am told to find the gradient of $\displaystyle f^-1(x)$ at the point ($\displaystyle \frac{7}{8}, \frac{1}{2}$)
$\displaystyle (f^{-1})'(7/8)=\frac{1}{f'(1/2)}=\ldots$

Edited: Corrected.

3. ## Re: inverse function problem

Originally Posted by FernandoRevilla
$\displaystyle (f^{-1})'(1/2)=\frac{1}{f'(7/8)}=\ldots$
I have already calculated dy/dx of f(x) which is 8x - 3x^2. If I use the y value of f(x) when x=7/8, then I get:

8(1/2) - 3(1/2)^2 = 4 - 3/4 = 13/4.

The reciprocal of this is 4/13 - which is correct answer.

4. ## Re: inverse function problem

Originally Posted by angypangy
I have already calculated dy/dx of f(x) which is 8x - 3x^2. If I use the y value of f(x) when x=7/8, then I get:

8(1/2) - 3(1/2)^2 = 4 - 3/4 = 13/4.

The reciprocal of this is 4/13 - which is correct answer.
You are right, I didn't check that $\displaystyle f(1/2)=7/8$ . I supposed (without computing) $\displaystyle f(7/8)=1/2$ .