Finding an inverse of a function?

Question

The function is f(x) =

1
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x^3 + 12x^2 + 48x + 43

I graphed it in my calculator, and it came out to be a parabola. Parabolas do not pass the horizontal line test to see if it is a one to one function or not. As far as I know, if a function is not one to one, then the inverse does not exist.

I want to find out another way to find out that there is no inverse to this function other than from graphing.
I know you can interchange x and y, but I don't know what to do after I completed that step. Can anyone point me in the right direction?
Thanks - phuoc

Answer 1

The denominator is the polynomial P(x) = x^3 + 12x^2 + 48x + 43. It's derivative is P'(x) = 3x^2 + 24x + 48 = 3(x^2 + 8x + 16) = 3(x +4)^2. So, P' has only one root at x = -4. Since P' is positive for x<>-4 and doesn't change its sign in a neighborhood of -4, it follows P is strictly increasing on R. At -4 we have an inflection but not an extreme point.

Since f(x) = 1/P(x), your f is not defined at x = -4. In (-oo, -4), f is negative and increases strictly in absolute value, so it's strictly decreasing. In (-4, oo), its clearly strictly decreasing. So, it's a bijection from (-oo, -4) U (-4, oo) to R, and therefore has an inverse.

But it's not possible to determine this inverse by means of elementary functions.

Answer 2

Define the domain and range for the function first
Or else it has no meaning talking about the inverse. =]

Answer 3

;)