Letfx-sin x-cos x- gx-x2-1- Thus gfx is invertible for x -A- -2- 0-B- -2- -C- -2- -4-D- -0- n-2-

The correct option is C By definition of composition of function , g(f(x))=(sinx+cosx)^2 1 ⇒ g(f(x))=sin2x We known sinx is bijiective only, when x ∈ [ π/2,π/ ...

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Standard XII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

Letfx=sin x+c...

Question

Let $f (x) = s i n x + c o s x$ , $g (x) = x^{2} - 1$ . Thus $g (f (x))$ is invertible for $x ϵ$

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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 known $s i n x$ is bijiective only, when $x \in [- \frac{π}{2}, \frac{π}{2}]$

Thus $g (x)$ is bijiective 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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Letf(x)=sinx+cosx , g(x)=x2−1 . Thus g(f(x)) is invertible for x ϵ

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

Let $f (x) = s i n x + c o s x$ , $g (x) = x^{2} - 1$ . Thus $g (f (x))$ is invertible for $x ϵ$