Question

Let $f\left(x\right)=\sqrt{2-x-x^2}$ and g(x) = cos x. Which of the following statements are true?
(I) Domain of $f\left(\right(g\left(x\right){)}^2)=$ Domain of f(g(x))
(II) Domain of f(g(x)) + g(f(x)) = Domain of g(f(x))
(III) Domain of f(g(x)) = Domain of g(f(x))
(IV) Domain of $g\left(\right(f\left(x\right){)}^3)=$ Domain of f(g(x))

• A. Only (I)
• B. Only (I) and (II)
• C. Only (III) and (IV)
• D. Only (I) and (IV)

Answer 1

Answer: (A)

Only (I)

Domain of $f\left(g\right(x\left)\right)$ is R

$\because f\left(g\right(x\left)\right)=2-\cos x-\cos x^2\ge 0$

$\Rightarrow (\cos x+2)(\cos x-1)\le 0$

$\Rightarrow$ $-2$ $\le$ $\cos x$ $\le$ $1$

$\therefore$ x $\in$ R

Domain of $g\left(f\right(x\left)\right)$ is [$-2$, $1$]

$\because$ In $\cos \sqrt{2-x-x^2}$

$2-x-x^2$ $\ge$ $0$

Domain of $f\left(\right(g\left(x\right){)}^2)$ is R

since $2$ - ${\cos }^{2}x$ - ${\cos }^{4}x$ $\ge$ $0$

(${\cos }^{2}x$ + $2$)(${\cos }^{2}x$ -$1$) $\le$ $0$

$-1$ $\le$ $\cos x$ $\le$ $1$

$\Rightarrow$ x$\in$ R

Domain of $g\left(\right(f\left(x\right){)}^3)$ is same as Domain of $g\left(f\right(x\left)\right)$

i.e. [$-2$,$1$]

From above discussion statement $\text{I}$ is true and statements $\text{II, III}$ and $\text{IV}$ are false.