Question

Let R⁺ be the set of all non-negative real numbers. If f: R⁺ → R⁺ and g: R⁺ → R⁺ are defined as f(x) = x² and g(x) = +√x, find fog and gof. Are they equal functions.

Answer 1

Given that f: R⁺ → R⁺ and g: R⁺ → R⁺

Therefore, fog: R⁺ → R⁺ and gof: R⁺ → R⁺

Domains of fog and gof are the same.

Let us find fog and gof also we have to check whether they are equal or not,

Consider that (fog)(x) = f(g(x))

= f(√x)

= √x²

= x

Now consider that (gof)(x) = g(f(x))

= g(x²)

= √x²

= x

Therefore, (fog)(x) = (gof)(x), ∀ x ∈ R⁺

Hence, fog = gof