If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?

Question

Philoid · Accepted Answer

We have, f (x) = sin x and g (x) = 2x.

Domain of f and g is R

Range of f = [ – 1,1], Range of g = R

∴ Range of f ⊂ Domain g and Range of g ⊂ Domain f

fog and gof both exist.

gof(x) = g(f(x)) = g(sin x)

⇒ gof(x) = 2sin x

fog(x) = f(g(x)) = f(2x) = sin 2x

∴ gof ≠ fog