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