Q. 13.9( 10 Votes )

# Find the domain of each of the following real valued functions of real variable:

i.

ii.

iii.

iv.

v.

Answer :

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 x^{2} – 9 = 0.

x^{2} – 9 = 0

⇒ x^{2} – 3^{2} = 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 x^{2} – 8x + 12 = 0.

x^{2} – 8x + 12 = 0

⇒ x^{2} – 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}

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