Find the domain of each of the following real valued functions of real

1.	Find the domain of each of the following real valued functions of real variable:(i) f (x) = 1/x(ii) f (x) = 1/(x - 7)(iii) f (x) = (3x - 2)/(x + 1)(iv) f (x) = (2x + 1)/(x2 - 9)(v) f (x) = (x2 + 2x + 1)/(x2 - 8x + 12)
Answer» (i) f(x) = 1/x As we know that, f(x) is defined for all real values of x, except for the case when x = 0. ∴ Domain of f = R – {0} (ii) f(x) = 1/(x - 7) As we know that, f (x) is defined for all real values of x, except for the case when x – 7 = 0 or x = 7. ∴ Domain of f = R – {7} (iii) f(x) = (3x - 2)/(x + 1) As we know that, f(x) is defined for all real values of x, except for the case when x + 1 = 0 or x = –1. ∴ Domain of f = R – {–1} (iv) f(x) = (2x + 1)/(x²- 9) As we know that, 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 ∴ Domain of f = R – {–3, 3} (v) f(x) = (x²+ 2x + 1)/(x²- 8x + 12) As we know that, 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 ∴ Domain of f = R – {2, 6}

Find the domain of each of the following real valued functions of real variable:(i) f (x) = 1/x(ii) f (x) = 1/(x - 7)(iii) f (x) = (3x - 2)/(x + 1)(iv) f (x) = (2x + 1)/(x2 - 9)(v) f (x) = (x2 + 2x + 1)/(x2 - 8x + 12)

Answer»

(i) f(x) = 1/x

As we know that, f(x) is defined for all real values of x, except for the case when x = 0.

∴ Domain of f = R – {0}

(ii) f(x) = 1/(x - 7)

As we know that, f (x) is defined for all real values of x, except for the case when x – 7 = 0 or x = 7.

∴ Domain of f = R – {7}

(iii) f(x) = (3x - 2)/(x + 1)

As we know that, f(x) is defined for all real values of x, except for the case when x + 1 = 0 or x = –1.

∴ Domain of f = R – {–1}

(iv) f(x) = (2x + 1)/(x²- 9)

As we know that, 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

∴ Domain of f = R – {–3, 3}

(v) f(x) = (x²+ 2x + 1)/(x²- 8x + 12)

As we know that, 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

∴ Domain of f = R – {2, 6}