Let R+ be the set of all non-negative real numbers. If f: R+ →�

1.	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» 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

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»

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