Find f + g, f – g, cf (c ∈ R, c ≠ 0), fg, 1/f and f/g in each of the following:
and
and
We have f(x) : [1, ∞) → R+ and g(x) : [–1, ∞) → R+ as real square root is defined only for non-negative numbers.
(a) f + g
We know (f + g)(x) = f(x) + g(x)
Domain of f + g = Domain of f ∩ Domain of g
⇒ Domain of f + g = [1, ∞) ∩ [–1, ∞)
∴ Domain of f + g = [1, ∞)
Thus, f + g : [1, ∞) → R is given by
(b) f – g
We know (f – g)(x) = f(x) – g(x)
Domain of f – g = Domain of f ∩ Domain of g
⇒ Domain of f – g = [1, ∞) ∩ [–1, ∞)
∴ Domain of f – g = [1, ∞)
Thus, f – g : [1, ∞) → R is given by
(c) cf (c ∈ R, c ≠ 0)
We know (cf)(x) = c × f(x)
Domain of cf = Domain of f
∴ Domain of cf = [1, ∞)
Thus, cf : [1, ∞) → R is given by
(d) fg
We know (fg)(x) = f(x)g(x)
Domain of fg = Domain of f ∩ Domain of g
⇒ Domain of fg = [1, ∞) ∩ [–1, ∞)
∴ Domain of fg = [1, ∞)
Thus, fg : [1, ∞) → R is given by
(e)
We know
Domain of = Domain of f
∴ Domain of = [1, ∞)
Observe that is also undefined when x – 1 = 0 or x = 1.
Thus, : (1, ∞) → R is given by
(f)
We know
Domain of = Domain of f ∩ Domain of g
⇒ Domain of = [1, ∞) ∩ [–1, ∞)
∴ Domain of = [1, ∞)
Thus, : [1, ∞) → R is given by