Finding the inverse of a function with two of the same variables

f(x) = 3x^{3} - x. If h(x) is the inverse of f(x) then h'(-2) =

Ok this is what I did:

1. Switched f(x) and replaced it with y, even though they're the same thing, but it's easier to see.

y = 3x^{3} - x

2. Switched the x and y variables in order to find the inverse.

x = 3y^{3} - y

3. Multiply both sides by natural log.

ln(x) = ln(3y^{3} - y)

4. Distributed the natural log on the other side.

ln(x) = ln(3y^{3}) - ln(y)

5. Use the log property, where subtracting two logs will mean dividing it.

ln(x) = ln(3y^{3})/ln(y)

6. Brought the exponent into the front.

ln(x) = 3ln(3y)/ln(y)

But then... I think I'm just experimenting and moving stuff around...

I don't think I'm really sure if I'm getting anywhere with this.

At first I stopped at step #2, and tried using natural log to see if it'll work.

But so far, I still have all 'y' on one side.

Anyone think they can help me out please?

Thanks :)

Re: Finding the inverse of a function with two of the same variables

Quote:

Originally Posted by

**Chaim** f(x) = 3x^{3} - x. If h(x) is the inverse of f(x) then h'(-2) =

You want to find $\displaystyle h'(-2)=\frac{1}{f'(h(-2))}$.

It is clear that $\displaystyle f(-1)=-2$ so $\displaystyle h(-2)=-1.$

Re: Finding the inverse of a function with two of the same variables

Quote:

Originally Posted by

**Plato** You want to find $\displaystyle h'(-2)=\frac{1}{f'(h(-2))}$.

It is clear that $\displaystyle f(-1)=-2$ so $\displaystyle h(-2)=-1.$

Oh wow, that makes more sense now.

So basically to find the inverse, you could just place under the denominator of 1 right?

Then after that you find out the derivative of f, which is 9x^{2} - 1

Then I plug in the -1, which makes the denominator = -10 right?

Then it would be -1/10?

Re: Finding the inverse of a function with two of the same variables

Quote:

Originally Posted by

**Chaim** Then after that you find out the derivative of f, which is 9x^{2} - 1

Then I plug in the -1, which makes the denominator = -10 right?

Then it would be -1/10?

Well no.

$\displaystyle f'(x)=9x^2-1$ so $\displaystyle f'(-1)=8.$

Re: Finding the inverse of a function with two of the same variables

Quote:

Originally Posted by

**Plato** Well no.

$\displaystyle f'(x)=27x^2-1$ so $\displaystyle f'(-1)=26.$

??? In the original post, the function [tex]f(x)= 3x^3- x[/quote] which has derivative $\displaystyle f'(x)= 9x^2- 1$, as Chaim said, not $\displaystyle 27x^2- 1$.