Question

Let $f\left(x\right)=\sqrt{2-x-x^2}$ and $g\left(x\right)=cosx$. Check given statement is true or not?

I. Domain of $f\left(g\right(x\left)\right)+g\left(f\right(x\left)\right)$ = Domain of $g\left(f\right(x\left)\right)$

Answer 1

$f\left(x\right)=\sqrt{2-x-x^2}$

$g\left(x\right)=\cos x$

Let $y_1=f\left(g\left(x\right)\right)=\sqrt{2-\cos x-{\cos }^{2}x}$

For existence of $y_1$, $2-cosx-co{s}^{2}x\ge 0$

$\Rightarrow {\cos }^{2}x+cosx-2\le 0$

$\Rightarrow \cos x\le 1$

$D\left(y_1\right)\Rightarrow x\in R$ ($\cos x\in [-1,1],\forall x\in R$)

Let $y_2,2-x-x^2\in R$

$\Rightarrow \forall x\in R,2-x-x^2\in R$

$D\left(y_2\right)=R$

$D\left(y_3\right)=D\left(g\left(f\left(x\right)\right)\right)=R\cup R$

$=R$

$=D\left(y_1\right)$

This statement is true.