<s>[INST] 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: $$ x =\frac{3\pm\sqrt{-7}}{2} $$
 [/INST] $$f(g(x)) = g(x)^2 + 1 = (3-x)^2 + 1$$
$$g(f(x)) = 3 - f(x) = 3 - x^2 - 1$$
Hence we need to find the points $x$ that satisfy
$$(3-x)^2 + 1 = 2 - x^2$$
We get that:
$$9 -6x+ x^2 + 1 -2 + x^2 = 0 \implies$$
$$2x^2 -6x + 8 = 0 \implies$$
$$x^2 - 3x + 4 = 0 \implies$$
$$...$$
you are correct.
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