Question

f(x) = 4x^3+2x^2+3x-1 find (f^-1)(-6) .....please explain

Answer 1

There are two ways to get the answer.
The first is to find the inverse function, and then substitute -6 for x to find f-1(-6).
Since this is a cubic, the inverse function is not simple to find. We can proceed to the second way when the inverse function is to be evaluated at a numerical value, namely -6.

The second way to evaluate the inverse function at a particular value (-6) is to find the solution to f(x)=-6, which is much easier to do.
If f(x0)=-6, then f-1(-6)=x0.
So solve for
4x^3+2x^2+3x-1=-6, or
4x^3+2x^2+3x+5=0
Descartes rule of signs tells us that there is at most 0 positive roots, and 3 negative roots.
Examination of the coefficients (4+3 = 2+5) tells us that there is a root at x=-1, which is our solution. It turns out (using a graph-plot) that the equation has only one real zero (x=-1).
Therefore
f-1(-6)=-1 or equivalently f(-1)=-6.