Question

How do you find the inverse of `F(x)= e^x+x^2+1`?

Answer 1

This function is not one-to-one so it has no inverse.

Explanation:

As `x->-oo`, `e^x->0` and `x^2->+oo`, hence `F(x)->+oo`

As `x->oo`, `e^x->+oo` and `x^2->+oo`, hence `F(x)->+oo`

So for sufficiently large values of `y` there are at least `2` values of `x` such that `F(x) = y`.

In fact, the graph of `F(x)` looks like this:

graph{e^x+x^2+1 [-20.33, 19.67, -5.36, 14.64]}

...similar to a parabola, but with a much steeper right hand side.