Find the domain of each of the following real valued functions of real variable:
i.
ii.
iii.
iv.
v.
i.
Clearly, f(x) is defined for all real values of x, except for the case when x = 0.
When x = 0, f(x) will be undefined as the division result will be indeterminate.
Thus, domain of f = R – {0}
ii.
Clearly, f(x) is defined for all real values of x, except for the case when x – 7 = 0 or x = 7.
When x = 7, f(x) will be undefined as the division result will be indeterminate.
Thus, domain of f = R – {7}
iii.
Clearly, f(x) is defined for all real values of x, except for the case when x + 1 = 0 or x = –1.
When x = –1, f(x) will be undefined as the division result will be indeterminate.
Thus, domain of f = R – {–1}
iv.
Clearly, f(x) is defined for all real values of x, except for the case when x2 – 9 = 0.
x2 – 9 = 0
⇒ x2 – 32 = 0
⇒ (x + 3)(x – 3) = 0
⇒ x + 3 = 0 or x – 3 = 0
⇒ x = ±3
When x = ±3, f(x) will be undefined as the division result will be indeterminate.
Thus, domain of f = R – {–3, 3}
v.
Clearly, f(x) is defined for all real values of x, except for the case when x2 – 8x + 12 = 0.
x2 – 8x + 12 = 0
⇒ x2 – 2x – 6x + 12 = 0
⇒ x(x – 2) – 6(x – 2) = 0
⇒ (x – 2)(x – 6) = 0
⇒ x – 2 = 0 or x – 6 = 0
⇒ x = 2 or 6
When x = 2 or 6, f(x) will be undefined as the division result will be indeterminate.
Thus, domain of f = R – {2, 6}