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

(fofof)(38)

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, ∞)

Clearly, range of f = [0, ∞) ⊄ Domain of (fof)

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

= {x: x ϵ [2, ∞) and ϵ [6, ∞)}

= {x: x ϵ [2, ∞) and ≥ 6}

= {x: x ϵ [2, ∞) and x – 2 ≥ 36}

= {x: x ϵ [2, ∞) and x ≥ 38}

= [38, ∞)

Now,

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

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

∴ fofof : [38, ∞) → R defined as

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

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

∴ fofof : [38, ∞) → R defined as

(fofof)(x) =

(fofof)(38) =

=