**Finding Domain of Rational Function as Union of Interval Notation :**

Here we are going to see, finding domain of rational function as union of interval notation

A rational function r is a function of the form r(x) = p(x) / q(x), where p and q are polynomials, with q ≠ 0.

The domain of a rational function is the set of real numbers where the expression defining the rational function makes sense.

Because division by 0 is not defined, the domain of a rational function p/q must exclude all zeros of q.

**Question 1 :**

Find the domain of the rational function r defined by

r(x) = (3x^{5} + x^{4} − 6x^{3} − 2) / (x^{2} − 9)

**Solution :**

The denominator of the expression above is 0 if x = 3 or x = −3. Thus unless stated otherwise, we would assume that the domain of r is the set of numbers other than 3 and −3.

In other words, the domain of r is (−∞,−3)∪(−3, 3)∪(3,∞).

**Question 2 :**

Find the domain of the rational function r defined by

r(x) = (5x^{3} − 12x^{2} + 13)/(x^{2} − 7)

**Solution :**

The denominator of the expression above is 0

x^{2} − 7 = 0

x^{2} = 7

x = √7

x = ± √7

if x = √7 or x = −√7. Thus unless stated otherwise, we would assume that the domain of r is the set of numbers other than √7 and −√7.

In other words, the domain of r is (−∞, −√7) U (−√7, √7) U (√7, ∞).

**Question 3 :**

Find the domain of the rational function r defined by

r(x) = (x^{5} + 3x^{4} - 6)/(2x^{2} − 5)

**Solution :**

The denominator of the expression above is 0

2x^{2} − 5 = 0

x^{2} = 5/2

x = ± √5/2

if x = √5/2 or x = −√5/2. Thus unless stated otherwise, we would assume that the domain of r is the set of numbers other than √5/2 and −√5/2.

In other words, the domain of r is (−∞, √5/2) U (−√5/2, √5/2) U (√5/2, ∞).

**Question 4 :**

Find the domain of the rational function r defined by

r(x) = (4x^{7} + 8x^{2} - 1)/(x^{2} - 2x − 6)

**Solution :**

The denominator of the expression above is 0

x^{2} - 2x − 6 = 0

x = [-b ± √(b^{2} - 4ac)] / 2a

x = [2 ± √(4 - 4(-6))] / 2(1)

x = [2 ± √28] / 2

x = (1 ± √7)

if x = 1 + √7 or x = 1 - √7. Thus unless stated otherwise, we would assume that the domain of r is the set of numbers other than 1 + √7 and 1 - √7.

In other words, the domain of r is (−∞, 1-√7) U (1-√7, 1+√7) U (1+√7, ∞).

After having gone through the stuff given above, we hope that the students would have understood "Finding Domain of Rational Function as Union of Interval Notation".

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