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

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Answer 1 · 2015-10-24T12:13:06

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

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.

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