A composite function problem The question is:
Suppose $$f(x) = x^2+1,$$ $$g(x) = 3-x.$$
Find the values for $x$ for such that $$(g\circ f)(x) = (f \circ g)(x).$$
I tried banging my head for one hour but my answer doesn't match the one given by the book which is $1/\sqrt{2}$ and $-1/\sqrt{2}$.
I think the answer given in the book is wrong because I even tried putting the given answer in $(g\circ f)(x)$ and $(f \circ g)(x)$ and the two don't match up.
My answer: $$ x =\frac{3\pm\sqrt{-7}}{2} $$

Given that $f(x) = x^2 + 1$ and $g(x) = 3 -x$, the composite function should look as follows for the LHS of the equation, hence,
$$3-(x^2 + 1)$$
RHS,
$$(3-x)^2 + 1$$
Thus,
$$ 3-(x^2 + 1) = (3-x)^2 + 1$$.
Simplifying we get,
$$2x^2-6x+8 = 0\\ x^2 - 3x + 4 = 0.$$
In order to find the value's of $x$ satisfying the equation we can use the quadratic formula, and you should get $\frac{3 \pm i\sqrt{7}}{2}$ where $i = \sqrt{-1}$. Meaning you are correct.
