Question

If $f:R\to R^{+}$ and $g:R^{+}\to R$ are such that $g\left(f\right(x\left)\right)=|\sin x|$ and $f\left(g\right(x\left)\right)=(\sin \sqrt{x}{)}^2$, then a possible choice for f and g is

• A. $f\left(x\right)=x^2,g\left(x\right)=\sin \sqrt{x}$
• B. $f\left(x\right)=\sin x,g\left(x\right)=|x|$
• C. $f\left(x\right)={\sin }^{2}x,g\left(x\right)=\sqrt{x}$
• D. $f\left(x\right)=x^2,g\left(x\right)=\sqrt{x}$

Answer 1

The correct option is A $f\left(x\right)=x^2,g\left(x\right)=\sin \sqrt{x}$

As f returns only positive values and takes any real number it can be mod or square hence option 2 is eliminated

For g only positive values has to be given and it will return any real number

so on trying the option only option $A$ and $C$ results in the given equation ie $g\left(f\right(x\left)\right)=|sinx|$

Now function g should take positive values and return real values which is satisfied by only option $A$ as in option $C$ square root of positive number will always result in positive number while $sin\sqrt{x}$ will give -ve real numbers as well.