Question

If $g(f(x)) = |\sin x|$ and $f(g(x)) = {(\sin \sqrt x )^2}$, then determine $f(x)$ and $g(x)$.
$f(x)$ and $g(x)$ are as follows:
a) $f(x) = {\sin ^2}x$ and $g(x) = \sqrt x$
b) $f(x) = \sin x$ and $g(x) = \left[ x \right]$
c) $f(x) = {x^2}$ and $g(x) = \sin \sqrt x$
d) $f$ and $g$ cannot be determined.

Answer 1

As we are given options here, we can go for checking option by option and then see which option satisfies the given conditions and the we can arrive at a conclusion. We know, $f(g(x))$ is basically substituting $g(x)$ in the place of $x$ in $f(x)$.Suppose, we have, $f(x) = x$, then $f(g(x)) = g(x)$. Also, we need to keep in mind that $\sqrt {{x^2}} = |x|$, not just $x$.

Complete answer: We are given $g(f(x)) = |\sin x|$ and $f(g(x)) = {(\sin \sqrt x )^2}$
In option a), we have $f(x) = {\sin ^2}x$ and $g(x) = \sqrt x$
So,
$\Rightarrow f(g(x)) = {\sin ^2}(g(x))$
$\Rightarrow f(g(x)) = {\sin ^2}(\sqrt x )$ as $(g(x) = \sqrt x )$
$\Rightarrow f(g(x)) = {(\sin \sqrt x )^2}$
And,
$g(f(x)) = \sqrt {f(x)}$
$\Rightarrow g(f(x)) = \sqrt {{{\sin }^2}x}$ as $f(x) = {\sin ^2}x$
$\Rightarrow g(f(x)) = |\sin x|$
We see that, option a) satisfies the given condition.
In option b), we have $f(x) = \sin x$ and $g(x) = \left[ x \right]$
So,
$f(g(x)) = \sin g(x)$
$\Rightarrow f(g(x)) = \sin \left[ x \right]$ as $g(x) = \left[ x \right]$
And,
$g(f(x)) = \left[ {f(x)} \right]$
$\Rightarrow g(f(x)) = \left[ {\sin x} \right]$ as $f(x) = \sin x$
We see that option b) does not satisfy the given condition and hence, option b) is discarded.
In option c), we have $f(x) = {x^2}$ and $g(x) = \sin \sqrt x$
So,
$f(g(x)) = {(g(x))^2}$
$\Rightarrow f(g(x)) = {(\sin \sqrt x )^2}$ as $g(x) = \sin \sqrt x$
$\Rightarrow f(g(x)) = {\sin ^2}\sqrt x$
And,
$g(f(x)) = \sin \sqrt {f(x)}$
$\Rightarrow g(f(x)) = \sin \sqrt {{x^2}}$ as $f(x) = {x^2}$
$\Rightarrow g(f(x)) = \sin |x|$
Hence, we see that option c) does not satisfy the given condition and hence option c) is discarded.
In option d), we have $f$ and $g$ cannot be determined
We already saw in option a) that we have such $f$ and $g$ which satisfy the given conditions and so we cannot just say that $f$ and $g$ cannot be determined.
Hence, option d) is discarded.
Therefore, the correct option is (a)
$f(x) = {\sin ^2}x$ and $g(x) = \sqrt x$

Note:
First of all, we need to be very careful with our concepts of trigonometry, like we have to keep in mind that ${(\sin x)^2} = {\sin ^2}x$, not ${(\sin x)^2} = {\sin ^2}{x^2}$. Also, we need to check every option even if we got our option a) correct or option b) correct. And, we need to take care while solving for the composition function i.e. we should take care that substitution is made everywhere in the function.