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

Answer 1

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

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

For existance of $y_1,2-\cos x-{\cos }^{2}x\ge 0$

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

$\Rightarrow (\cos x-1)\le 0$

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

$y_2=g\left(f\left(x\right)\right)=\cos (2-x-x^2)$

For existance of $y_2,2-x-x^2\in R$, we know that,

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

$D\left(y_1\right)=D\left(y_2\right)$

This statement is true.