Question

If f: R⁺ → R⁺ and g : R⁺ → R⁺, defined as f(x) = x², g(x) = √x, then find gof and fog wheather are they equivalent ?

Answer 1

Given, f: R⁺ → R⁺, f(x) = x²

g : R⁺ → R⁺, g(x)= √x

Then, (fog): R⁺ → R⁺ and (gof): R⁺ → R⁺ are defined

∴ (gof)(x) = g[f(x)]

= g(x²) = √x² = x

(fog)(x) = f(g(x)]

= f(√x)= (√5)² = x

Based on above, (gof) and (fog) are of same domain and co-domain.

(fog)(x) = (gof)(x) ∀ x ∈ R⁺

So, (fog) and (gof) are equivalent functions.