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 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}


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