How do I find the inverse of x^3/x^2+1?

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In summary, the problem is x^3/x^2+1 and the goal is to find the inverse by isolating x. The conversation discusses a similar problem that involved expressing the statement twice and canceling terms to isolate x, but this method does not work for this problem. The conversation concludes with the acknowledgement that solving this problem will require using Cardano's equation to find the solutions of the cubic equation.
  • #1
Bardagath
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Hello,

I have been working on this problem for a while now and can't seem to get it into a forum which would isolate x, which would then allow me to sub y and find the inverse.

The problem is x^3/x^2+1

Can anyone help me with this? I am utterly stuck

I have dealt with a similar problem before... x+1/x-1... the trick for that was to express the statement twice, once with x-1 in the numerator and x+2 in the numerator... which would cancel the first statement and isolate x etc.

However, this problem we have a cube and a square in the numerator and the denomator.

Can someone please help?

Thanks
 
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  • #2
Bardagath said:
Hello,

I have been working on this problem for a while now and can't seem to get it into a forum which would isolate x, which would then allow me to sub y and find the inverse.

The problem is x^3/x^2+1
Presumably you mean f(x) = x3/(x2 + 1). What you wrote would be interpreted as (x3/x2) + 1 = x + 1.
Bardagath said:
Can anyone help me with this? I am utterly stuck

I have dealt with a similar problem before... x+1/x-1...
The similar problem was y = (x + 1)/(x - 1), I'm pretty sure.
Bardagath said:
the trick for that was to express the statement twice, once with x-1 in the numerator and x+2 in the numerator... which would cancel the first statement and isolate x etc.
(x + 1)/(x - 1) = (x - 1 + 2)/(x - 1) = (x - 1)/(x - 1) + 2/(x - 1) = 1 + 2/(x - 1). Is this what you're talking about? If so, it's fairly easy to solve the equation y = 1 + 2/(x - 1) for x.
Bardagath said:
However, this problem we have a cube and a square in the numerator and the denomator.

Can someone please help?

Thanks
Solving the equation y = x3/(x2 + 1) for x is possible, but pretty difficult.

Multiply both sides by x2 + 1:
y(x2 + 1) = x3

Expand the left side and bring all terms to the left side:
yx2 + y - x3 = 0

Rearrange by powers of x:
-x3 + yx2 + y = 0

Solving for x amounts to finding the solutions of this cubic equation, a technique that has been around for a long time, but isn't usually taught.
 
  • #3
Thanks so much,

I was afraid I'd be left in this form. Oh well, must use Cardano's equation now...

Thankyou very much
 

1. What is the inverse of X^3/X^2+1?

The inverse of X^3/X^2+1 is (X^2+1)/X^3.

2. How do you find the inverse of X^3/X^2+1?

To find the inverse of X^3/X^2+1, you can switch the numerator and denominator to get (X^2+1)/X^3.

3. Is the inverse of X^3/X^2+1 the same as the reciprocal?

Yes, the inverse of X^3/X^2+1 is the same as the reciprocal because they both involve switching the numerator and denominator.

4. Can the inverse of X^3/X^2+1 be simplified?

Yes, the inverse of X^3/X^2+1 can be simplified to (X^2+1)/X^3.

5. How is the inverse of X^3/X^2+1 related to the original function?

The inverse of X^3/X^2+1 is the function that "undoes" the original function. In other words, if you apply the original function and then the inverse function, you will get back to your original input. In this case, (X^2+1)/X^3 is the inverse function of X^3/X^2+1.

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