If you have a quadratic where "a" and "b" values are given, how do you find the values of c so that the quadratic intersects with its inverse?

Question

I have a question where it is asking to find the values of a for which the graph of $f(x) = x^2 + 2x + a$ will intersect with its inverse. I solved it to the point where $a = -x(x-3)$, but how do you solve for $a$?

Answer 1

So $f(x) = x^2+2x+a = (x+1)^2+(a-1)$ and to find the inverse, we have $$ \begin{split} x &= (y+1)^2+(a-1) \\ (y+1)^2 &= x-a+1 \\ y &= -1 \pm \sqrt{x-a+1} \end{split} $$ Thus, $f^{-1}(x) = -1 \pm \sqrt{x-a+1}$.

Where will $f(x)$ and $f^{-1}(x)$ intersect?