$(f \circ g)(x) = \sin (x^{1/2})^2, \; (g \circ f)(x) = | \sin x |$

Question

If

$(f \circ g)(x) = \sin (\sqrt{x})^2$

$(g \circ f)(x) = | \sin x |$

Find $f(x)$ , $g(x)$.

I'm told there are 2 solutions.

I do not have an idea of how to approach these questions. Would you please also give the required line of thought. Thank you!!

Answer 1

The absolute value must have come from $|x|=(x^2)^{1/2}$ since these powers appear in the other function. Also, it seems that the square is always outside the sine whereas the square root is inside in $f\circ g$ but outside in $g\circ f$. This suggests that the square and the sine are in one function and the square root is in the other. Indeed, one example is given by $g(t)=t^{1/2}$ and $f(t)=\sin^2(t)$.

Answer 2

Hint:

The absolute value doesn't come out of nowhere, so we can rewrite $g\circ f=\sqrt{\sin^2(x)}$.

Can you find $f(x)$ and $g(x)$ now?