KVPY SX SB Exam 2013 Question Paper

Domain of f(g(x)) is R
∵ 2 - cosx - cos${}^2$x ≥ 0
(cos x + 2)(cosx - 1) ≤ 0
⇒ - 2 ≤ cos x ≤ 1
x ∈ R
Domain of g(f(x)) is [-2, 1]
$\because \cos \left(\sqrt{2-\mathit{x}-{\mathit{x}}^2}\right)$
$2-\mathit{x}-{\mathit{x}}^2\ge 0$
Domain of $\mathrm{f}\left(\mathrm{g}{\left(\mathit{x}\right)}^2\right)$ is R
∵ 2 - cos${}^2$x - cos${}^4$x ≥ 0
(cos${}^2$x + 2)(cos${}^2$x - 1) ≤ 0
-1 ≤ cos x ≤ 1
x ∈ R
Domain of $\mathrm{g}\left({\mathrm{f}}^3\left(\mathrm{x}\right)\right)$ is Domain of g(f(x)
i.e., [-2, 1]