Let f be a real function given by . Find each of the following:

f²

Also, show that fof ≠ f².

We have, f(x) =

Clearly, domain of f = [2, ∞] and range of f = [0, ∞)

We observe that range of f is not a subset of domain of f

∴ Domain of (fof) = {x: x ϵ Domain of f and f(x) ϵ Domain of f}

= {x: x ϵ [2, ∞) and ϵ [2, ∞)}

= {x: x ϵ [2, ∞) and ≥ 2}

= {x: x ϵ [2, ∞) and x – 2 ≥ 4}

= {x: x ϵ [2, ∞) and x ≥ 6}

= [6, ∞)

Now,

(fof)(x) = f(f(x)) = f =

∴ fof: [6, ∞) → R defined as

(fof)(x) =

f²(x) = [f(x)]² = = x – 2

∴ f²: [2, ∞) → R defined as

f²(x) = x – 2

∴ fof ≠ f²