Finding the inverse of this function explicitly

[Nano-Passion]

Homework Statement

Find the inverse of this function
$$f(x) = x^3+2x-1$$

Homework Equations

The Attempt at a Solution

I'm not able to state the function explicitly.
$$f(x)=x^3+2x-1$$
$$y=x^3+2x-1$$
(switch all x's and y's to find inverse)
$$x=y^3+2y-1$$
$$y^3+2y=x-1$$
$$y(y^2+1)=x-1$$
$$y=\frac{x-1}{y^2+2}$$

 

[eumyang]

I'm not able to state the function explicitly.
$$f(x)=x^3+2x-1$$
$$y=x^3+2x-1$$
(switch all x's and y's to find inverse)
$$x=y^3+2y-1$$
$$y^3+2y=x-1$$

Move everything to the left side:
$$y^3+2y+(1-x)=0$$
You have a cubic equation in y, if you treat x as a constant. You can try solving it using Cardano's method, but it's going to be VERY messy. Are you sure you copied the problem correctly? Is it in a textbook?

 

[Dick]

You could use the cubic formula to solve for an inverse, but that's extremely complicated. Are you sure you need an explicit inverse? Or do you just have to show that an inverse exists?

 

[Nano-Passion]

Move everything to the left side:
$$y^3+2y+(1-x)=0$$
You have a cubic equation in y, if you treat x as a constant. You can try solving it using Cardano's method, but it's going to be VERY messy. Are you sure you copied the problem correctly? Is it in a textbook?

I'm certain I copied it right. It didn't ask me to find it the inverse explicitly though, verify that f had an inverse. It was actually part of a calc problem but this portion of the problem isn't calculus based.

You could use the cubic formula to solve for an inverse, but that's extremely complicated. Are you sure you need an explicit inverse? Or do you just have to show that an inverse exists?

I needed to show that an inverse exists.

 

[eumyang]

I'm certain I copied it right. It didn't ask me to find it the inverse explicitly though, verify that f had an inverse.

Then why do you have "explicitly" in the title of the thread? I'd change the title if I were you.

 

[Dick]

I'm certain I copied it right. It didn't ask me to find it the inverse explicitly though, verify that f had an inverse. It was actually part of a calc problem but this portion of the problem isn't calculus based.



I needed to show that an inverse exists.

Use some calculus, look at the derivative. What does that tell you about the function f(x)=x^3+2x-1?

 

[Mark44]

I'm certain I copied it right. It didn't ask me to find it the inverse explicitly though, verify that f had an inverse. It was actually part of a calc problem but this portion of the problem isn't calculus based.

Then why do you have "explicitly" in the title of the thread?

In fact, the whole title is misleading. The problem isn't asking you to find the inverse - just verify that the function has an inverse. Those are two different things.

 

[Nano-Passion]

Use some calculus, look at the derivative. What does that tell you about the function f(x)=x^3+2x-1?

That the function is differentiable in x^3 & 2x & -1 ?

In fact, the whole title is misleading. The problem isn't asking you to find the inverse - just verify that the function has an inverse. Those are two different things.

Well how would you verify it is an inverse otherwise?

 

[Mark44]

If the function given in the first post is increasing everywhere or decreasing everywhere, it is one-to-one, which guarantees that it has an inverse. The derivative of the function can tell you where the function is increasing or decreasing. That's where Dick was steering you.

 

[Ray Vickson]

If the function given in the first post is increasing everywhere or decreasing everywhere, it is one-to-one, which guarantees that it has an inverse. The derivative of the function can tell you where the function is increasing or decreasing. That's where Dick was steering you.

But (and we don't know this): does the OP have access yet to the required theorem about inverses of monotone functions, or is he/she being asked to somehow prove it in this special case?

RGV

 

[Nano-Passion]

But (and we don't know this): does the OP have access yet to the required theorem about inverses of monotone functions, or is he/she being asked to somehow prove it in this special case?

RGV

Nope, never heard of the term actually.

 

[Dick]

Nope, never heard of the term actually.

You could look it up. And there's no heavy duty theorem involved here. What properties does a function need to be invertible? Look up "horizontal line test" if you need it spelled out. What might a derivative of the function have to do with that? You should really think about this some more.

 

[willem2]

You could use the intermediate value theorem to prove that $x^3 + 2x + a =0$ must have at least one root, and then prove that it can't have 2 or 3 roots.

- use the fact that if a polynomial has a root r, it is divisible by (x-r) to prove
that the polymial can be factored in the form $(x-r_1)(x-r_2)(x-r_3)$ in both cases.

- Then prove that the product $(x-r_1)(x-r_2)(x-r_3)$ can't be a cubic with a 0 quadratic term and a positive linear term.