Let f (x) = x2 and g(x) = sin x for all x ∈ R. Then the set of all x satisfying (f o g o g o f ) (x) = (g o g o f) (x), where (f o g) (x) = f (g(x)), is

Engineering

Mathematics

Composition of Functions

Question

Let f (x) = x² and g(x) = sin x for all x ∈ R. Then the set of all x satisfying

(f o g o g o f ) (x) = (g o g o f) (x), where (f o g) (x) = f (g(x)), is

$\pm \sqrt{nπ}, n \in {0, 1, 2, . .......}$

$\frac{π}{2} + 2 nπ, n \in {. .......... - 2, - 1, 0, 1, 2, . ......}$

2n $π$ , n $\in$ {......., – 2, – 1, 0, 1, 2, .......}

$\pm \sqrt{nπ}, n \in {1, 2, . ..........}$

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gof(x) = gf(x) = g(x2) = sin x²

Go (gof(x)) = g(sin x²) = sin (sin x²)

Fo(gogof(x)) = f(sin (sin x²)) = (sin(sin x²))²

$∴$ (sin (sin x²))² = sin (sin x²)

Sin (sin x²) (sin (sin x²) –1) = 0

Sin (sin x²) = 0 or sin (sin x²) = 1

Sin x² = np sin x² = 2np + $\frac{π}{2}$

At n = 0 At n = 0

Sin x² = 0 $\sin x^{2} = \frac{π}{2}$

x² = n $π$ Not possible

$x - \pm \sqrt{nπ};$ n $\in$ {0, 1, 2, …..}