HOW TO FIND DOMAIN OF A RATIONAL FUNCTION

Find the domain of the rational function.

Example 1 :

r(x) = (3x⁵ + x⁴ - 6x³ - 2)/(x² - 9) 

Solution :

Equate the denominator to zero.

x² - 9 = 0

x² - 3² = 0

Using algebraic identity a² - b² = (a + b)(a - b),

(x + 3)(x - 3) = 0

x + 3 = 0   or   x - 3 = 0

x = -3   or   3

The domain is all real values except -3 and 3.

In interval notation,

(-∞, -3)U(-3, 3)U(3, +∞)

Example 2 :

r(x) = (5x³ − 12x² + 13)/(x² −  7)

Solution :

Equate the denominator to zero.

x² - 7 = 0

x² - (√7)² = 0

Using algebraic identity a² - b² = (a + b)(a - b),

(x + √7)(x - √7) = 0

x + √7 = 0   or   x - √7 = 0

x = -√7   or   √7

The domain is all real values except -√7 and √7.

In interval notation,

(-∞, -√7)U(-√7, √7)U(√7, +∞)

Example 3 :

r(x) = (x⁵ + 3x⁴ - 6)/(2x² − 5)

Solution :

Equate the denominator to zero.

2x² - 5 = 0

x² = 5/2

x = ±√(5/2)

The domain is all real values except ±√(5/2).

In interval notation,

(-∞, -√5/2)U(-√5/2,  √5/2)U(√5/2, +∞)

Example 4 :

r(x) = (4x⁷ + 8x² - 1)/(x² - 2x − 6)

Solution :

Equate the denominator to zero.

x² - 2x - 6 = 0

The above quadratic equation can not be solved by factoring.

Use quadratic formula to solve the equation.

Comparing ax² + bx + c = 0 and x² - 2x - 6 = 0,

a = 1, b = -2 and c = -6

Quadratic formula :

Substitute a = 1, b = -2 and c = -6.

The domain is all real values except 1 ± √7.

In interval notation,

(-∞, 1 - √7)U(1 - √7,  1 + √7)U(1 + √7, +∞)

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