Compute the inverse function of the following polynomial on [ 0 , 1 ]? f ( x ) = α x 3 − 2 α x 2 + ( α + 1 ) x(where α is within ] 0 , 3 [)

Question

Compute the inverse function of the following polynomial on   [  0  ,  1  ]?  f  (  x  )  =  α      x    3    −  2  α      x    2    +  (  α  +  1  )  x(where   α is within   ]  0  ,  3  [)

Alisa Gilmore · Accepted Answer

J.M. is right - in general there is no inverse function, because your function is not injective.assume that the domain is real numbers.However, we can check when it is not injective. If it is monotonic, then it is injective (since it is a polynomial so it does not have a constant fragments unless it is a constant function - a polynomial of degree 0). And if a polynomial is monotonic, then its derivative is nonpositive or nonnegative. On the other hand if its derivative takes values of both signs, then it is not injective (it first go down from some value, then there is minimum, and then it goes up - a U-shaped function is obviously not injective; the same for first going up and then down).So we calculate the derivative   3  α      x    2    −  4  α  x  +  α  +  1. It takes values of both signs if it has two distinct roots, so when the   Δ is positive.  Δ  =  16      α    2    −  4  (  α  +  1  )  3  α  =  16      α    2    −  12      α    2    −  12  α  =  4      α    2    −  12  α  =  4  α  (  α  −  3  )so it is positive for all   α  ∈  [  −  ∞  ,  0  )  ∪  (  3  ,  ∞  ]So the task have sense only for   α  ∈  [  0  ,  3  ] (assuming   x is real, otherwise [in the case of       C  ] the function is not injective for   α  ≠  0).When you want to calculate       f          −      1        (  y  ) you want to find   x such that   y  =  f  (  x  ), so you&#039;re solving the equation:  α      x    3    −  2  α      x    2    +  (  α  +  1  )  x  −  y  =  0.

Compute the inverse function of the following polynomial on [ 0 , 1 ] ? f (

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