Let f x -sin x -cos x - g x - x 2-1- Thus g f x is invertible for x -A- -2- -B- -2- -4-C- -0- -2-D- -2- 0-

The correct option is B By definition of composition of function, g(f(x))=(sin x+cosx)^2 1⇒ g(f(x))=sin2x We know sin x is bijective only, when x∈ [ π/2,π/2] ...

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Standard XII

Physics

Definite Integrals

Let f x =sin ...

Question

Let f(x) = sin x + cos x, $g (x) = x^{2} - 1$ . Thus g(f(x)) is invertible for x $\in$

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Solution

The correct option is C
By definition of composition of function,

$g (f (x)) = (s i n x + c o s x)^{2} - 1 \Rightarrow g (f (x)) = s i n 2 x$

We know sin x is bijective only, when $x \in [- \frac{π}{2}, \frac{π}{2}]$

Thus g(x) is bijective if $- \frac{π}{2} \leq 2 x \leq \frac{π}{2} \Rightarrow - \frac{π}{4} \leq x \leq \frac{π}{4} .$

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Let f(x) = sin x + cos x, g(x)=x2−1. Thus g(f(x)) is invertible for x∈

The correct option is C By definition of composition of function, g(f(x))=(sinx+cosx)2−1⇒g(f(x))=sin2x We know sin x is bijective only, when x∈[−π2,π2] Thus g(x) is bijective if −π2≤2x≤π2 ⇒−π4≤x≤π4.

Let f(x) = sin x + cos x, $g (x) = x^{2} - 1$ . Thus g(f(x)) is invertible for x $\in$