FINDING DOMAIN OF RATIONAL FUNCTIONS

State the domain as interval notation.

Problem 1 :

Solution :

To find the domain of the rational function given above, we equate the denominator to 0.

x - 5 = 0

x = 5

5 is value which will make the function as undefined, so this value should be excluded from the domain.

Domain is R - {5}

Required interval notation is (-∞, 5) U (5, ∞)

Problem 2 :

Solution :

Factoring the denominator,

x² - 3x - 108

x² - 12x + 9x - 108

= x(x - 12) + 9(x - 12)

= (x + 9)(x - 12)

Equating the factors to 0, we get x = -9 and x = 12

The values -9 and 12 will make the function as 0. These values should be excluded from domain.

Domain as interval notation is (-∞, -9) U (-9, 12) U (12, ∞)

Problem 3 :

Solution :

Factoring the denominator,

x² - 3x - 28

x² - 7x + 4x - 28

= x(x - 7) + 4(x - 7)

= (x + 4)(x - 7)

Equating the factors to 0, we get x = -4 and x = 7

The values -4 and 7 will make the function as 0. These values should be excluded from domain.

Domain as interval notation is (-∞, -4) U (-4, 7) U (7, ∞)

Problem 4 :

Solution :

= 9x² - 6x + 1

= 9x² - 3x - 3x + 1

= 3x (3x - 1) - 1(3x - 1)

= (3x - 1)(3x - 1)

Equating each factor to 0, we get

x = 1/3 and 1/3

1/3 will make the denominator 0. So, this alone should be excluded 

Domain as interval notation is (-∞, 1/3) U (1/3, ∞)

Problem 5 :

Solution :

= 2x² - 5x + 3

= 2x² - 2x - 3x + 3

= 2x (x - 1) - 3(x - 1)

= (2x - 3)(x - 1)

Equating each factor to 0, we get

x = 3/2 and 1

3/2 and 1 will make the denominator 0. So, these two values  should be excluded 

Domain as interval notation is (-∞, 3/2) U (3/2, 1) U(1, ∞).

Problem 6 :

Solution :

x² + 5 = 0

x² = -5

x = √-5 (not defined)

No values will make the denominator as 0.

So, domain is (-∞, ∞)

Problem 7 :

Solution :

= x² + 3x - 10

= x² + 5x - 2x - 10

= x(x + 5) - 2(x + 5)

= (x - 2)(x + 5)

Equating each factor to 0, we get

x = 2 and x = -5

2 and -5 will make the denominator 0. So, these two values  should be excluded 

Domain as interval notation is (-∞, -5) U (-5, 2) U (2, ∞).

Problem 8 :

Solution :

= x² - 3x

= x(x - 3)

Equating each factor to 0, we get

x = 0 and x = 3

0 and 3 will make the denominator 0. So, these two values  should be excluded 

Domain as interval notation is (-∞, 0) U (0, 3) U (3, ∞).

Problem 9 :

Solution :

√(2x -10) > 0

2x - 10 > 0

2x > 10

x > 5

5 is the value should be excluded from the domain.

So, the domain is (5, ∞)

Problem 10 :

Solution :

√(x² - 100) > 0

x² - 100 > 0

x²  > 100

Take square root on both side

x > ±10

-10 and 10 should be excluded from the domain.

So, the domain is (-∞, -10) U (10, ∞)