Given f(x) = x + 1 and g(x) = 2x – 3

Clearly, both f(x) and g(x) exist for all real values of x.

Hence, Domain of f = Domain of g = R

Range of f = Range of g = R

i. f + g

We know (f + g)(x) = f(x) + g(x)

⇒ (f + g)(x) = x + 1 + 2x – 3

∴ (f + g)(x) = 3x – 2

Domain of f + g = Domain of f ∩ Domain of g

⇒ Domain of f + g = R ∩ R

∴ Domain of f + g = R

Thus, f + g : R → R is given by (f + g)(x) = 3x – 2

ii. f – g

We know (f – g)(x) = f(x) – g(x)

⇒ (f – g)(x) = x + 1 – (2x – 3)

⇒ (f – g)(x) = x + 1 – 2x + 3

∴ (f – g)(x) = –x + 4

Domain of f – g = Domain of f ∩ Domain of g

⇒ Domain of f – g = R ∩ R

∴ Domain of f – g = R

Thus, f – g : R → R is given by (f – g)(x) = –x + 4

iii.

We know

Clearly, is defined for all real values of x, except for the case when 2x – 3 = 0 or.

When, will be undefined as the division result will be indeterminate.

Thus, domain of = R –

Thus, : R – → R is given by