Domain of a Rational Function

Domain of a Rational Function. Practice questions to deepen understanding of the domain of a rational function. Online math practice with full solutions and step-by-step explanations.

Domain of a Rational Function — 30 questions: denominator nonzero, factoring, difference of squares, trinomials. Detailed explanations.

Topics covered:

Basic questions (1-5): simple denominators
Difference of squares (3, 4, 11, 14, 27)
Perfect squares (12, 16, 24, 26, 30)
Trinomials (7, 9, 17, 21, 22)
Common factor extraction (8, 15, 23)
Denominators that never vanish (6, 20)
Pre-factored products (5, 19)
Powers of x (18)
Denominators with fractions (25)
Sum of cubes (28)
Complex cases (29, 30)

Each question includes:

All quote marks fixed ('')
Detailed 5-7 step explanations
Full factorization of the denominator
Comprehensive numerical checks
Tables with dir="ltr"
Important notes on common mistakes
Clear distinction between numerator and denominator
No visual examples

30 questions

Question 1

3.33 pts

Find the domain of \(f(x)=\dfrac{1}{x-3}\).

\(x\neq3\)

\(x>3\)

\(x\geq3\)

All real numbers

Explanation:

Solution: Denominator x-3≠0.

Question 2

3.33 pts

Find the domain of \(f(x)=\dfrac{x+1}{x+5}\).

\(x\neq-5\)

\(x\neq-1\)

\(x\neq5\)

\(x\neq-1,\,x\neq5\)

Explanation:

Solution: x+5≠0.

Question 3

3.33 pts

Find the domain of \(f(x)=\dfrac{1}{x^2-4}\).

\(x\neq2,\,x\neq-2\)

\(x\neq4\)

\(x\neq2\)

\(x>2\)

Explanation:

Solution: x²-4=(x-2)(x+2)≠0.

Question 4

3.33 pts

Find the domain of \(f(x)=\dfrac{2x+3}{x^2-9}\).

\(x\neq3,\,x\neq-3\)

\(x\neq9\)

\(x\neq-\frac{3}{2}\)

\(x\neq3\)

Explanation:

Solution: x²-9=(x-3)(x+3)≠0.

Question 5

3.33 pts

Find the domain of \(f(x)=\dfrac{1}{(x-1)(x+2)}\).

\(x\neq1,\,x\neq-2\)

\(x\neq1\)

\(x\neq-2\)

\(x\neq2\)

Explanation:

Solution: Each factor ≠ 0.

Question 6

3.33 pts

Find the domain of \(f(x)=\dfrac{x}{x^2+4}\).

All real numbers \((\mathbb{R})\)

\(x\neq2\)

\(x\neq4\)

\(x>0\)

Explanation:

Solution: x²+4≥4>0 — never zero.

Question 7

3.33 pts

Find the domain of \(f(x)=\dfrac{x-2}{x^2-5x+6}\).

\(x\neq2,\,x\neq3\)

\(x\neq6\)

\(x\neq2\)

\(x\neq-2,\,x\neq-3\)

Explanation:

Solution: x²-5x+6=(x-2)(x-3)≠0.

Question 8

3.33 pts

Find the domain of \(f(x)=\dfrac{1}{x^2-x}\).

\(x\neq0,\,x\neq1\)

\(x\neq1\)

\(x\neq0\)

\(x\neq-1,\,x\neq1\)

Explanation:

Solution: x²-x=x(x-1)≠0.

Question 9

3.33 pts

Find the domain of \(f(x)=\dfrac{3x+1}{x^2+x-6}\).

\(x\neq2,\,x\neq-3\)

\(x\neq6\)

\(x\neq-1\)

\(x\neq2,\,x\neq3\)

Explanation:

Solution: x²+x-6=(x-2)(x+3)≠0.

Question 10

3.33 pts

Find the domain of \(f(x)=\dfrac{x+5}{2x-8}\).

\(x\neq4\)

\(x\neq-5\)

\(x\neq8\)

\(x\neq2\)

Explanation:

Solution: 2x-8=2(x-4)≠0.

Question 11

3.33 pts

Find the domain of \(f(x)=\dfrac{x^2+1}{x^2-16}\).

\(x\neq4,\,x\neq-4\)

\(x\neq16\)

\(x\neq4\)

All real numbers

Explanation:

Solution: x²-16=(x-4)(x+4)≠0.

Question 12

3.33 pts

Find the domain of \(f(x)=\dfrac{1}{x^2+6x+9}\).

\(x\neq-3\)

\(x\neq3\)

\(x\neq-3,\,x\neq3\)

All real numbers

Explanation:

Solution: x²+6x+9=(x+3)²≠0.

Question 13

3.33 pts

Find the domain of \(f(x)=\dfrac{2x-1}{3x+6}\).

\(x\neq-2\)

\(x\neq6\)

\(x\neq\frac{1}{2}\)

\(x\neq2\)

Explanation:

Solution: 3x+6=3(x+2)≠0.

Question 14

3.33 pts

Find the domain of \(f(x)=\dfrac{x^3}{x^2-1}\).

\(x\neq1,\,x\neq-1\)

\(x\neq1\)

\(x\neq0\)

All real numbers

Explanation:

Solution: x²-1=(x-1)(x+1)≠0.

Question 15

3.33 pts

Find the domain of \(f(x)=\dfrac{5}{x^2+2x}\).

\(x\neq0,\,x\neq-2\)

\(x\neq2\)

\(x\neq0\)

\(x\neq-2\)

Explanation:

Solution: x²+2x=x(x+2)≠0.

Question 16

3.33 pts

Find the domain of \(f(x)=\dfrac{x^2-4}{x^2-4x+4}\).

\(x\neq2\)

\(x\neq2,\,x\neq-2\)

\(x\neq4\)

All real numbers

Explanation:

Solution: x²-4x+4=(x-2)²≠0.

Question 17

3.33 pts

Find the domain of \(f(x)=\dfrac{1}{x^2-7x+12}\).

\(x\neq3,\,x\neq4\)

\(x\neq12\)

\(x\neq7\)

\(x\neq-3,\,x\neq-4\)

Explanation:

Solution: x²-7x+12=(x-3)(x-4)≠0.

Question 18

3.33 pts

Find the domain of \(f(x)=\dfrac{x+3}{x^2}\).

\(x\neq0\)

\(x\neq-3\)

All real numbers

\(x\neq3\)

Explanation:

Solution: x²=0 only at x=0.

Question 19

3.33 pts

Find the domain of \(f(x)=\dfrac{x-1}{(x+2)(x-3)}\).

\(x\neq-2,\,x\neq3\)

\(x\neq1\)

\(x\neq2,\,x\neq3\)

\(x\neq-2,\,x\neq-3\)

Explanation:

Solution: Each factor ≠ 0.

Question 20

3.33 pts

Find the domain of \(f(x)=\dfrac{2}{x^2+9}\).

All real numbers \((\mathbb{R})\)

\(x\neq3,\,x\neq-3\)

\(x\neq9\)

\(x\neq0\)

Explanation:

Solution: x²+9≥9>0 — never zero.

Question 21

3.33 pts

Find the domain of \(f(x)=\dfrac{x}{x^2-3x-10}\).

\(x\neq5,\,x\neq-2\)

\(x\neq10\)

\(x\neq3\)

\(x\neq2,\,x\neq5\)

Explanation:

Solution: x²-3x-10=(x-5)(x+2)≠0.

Question 22

3.33 pts

Find the domain of \(f(x)=\dfrac{x^2+5x+6}{x^2-9}\).

\(x\neq3,\,x\neq-3\)

\(x\neq-2,\,x\neq-3\)

\(x\neq6,\,x\neq-6\)

\(x\neq9\)

Explanation:

Solution: x²-9=(x-3)(x+3)≠0.

Question 23

3.33 pts

Find the domain of \(f(x)=\dfrac{1}{x^3-x}\).

\(x\neq0,\,x\neq1,\,x\neq-1\)

\(x\neq0\)

\(x\neq1\)

\(x\neq-1\)

Explanation:

Solution: x³-x=x(x-1)(x+1)≠0.

Question 24

3.33 pts

Find the domain of \(f(x)=\dfrac{x-4}{x^2-8x+16}\).

\(x\neq4\)

\(x\neq4,\,x\neq-4\)

\(x\neq8,\,x\neq0\)

\(x\neq-4\)

Explanation:

Solution: x²-8x+16=(x-4)²≠0.

Question 25

3.33 pts

Find the domain of \(f(x)=\dfrac{x^2}{4x^2-1}\).

\(x\neq\dfrac{1}{2},\,x\neq-\dfrac{1}{2}\)

\(x\neq0\)

\(x\neq4,\,x\neq-4\)

\(x\neq1\)

Explanation:

Solution: 4x²-1=(2x-1)(2x+1)≠0.

Question 26

3.33 pts

Find the domain of \(f(x)=\dfrac{1}{x^2+4x+4}\).

\(x\neq-2\)

\(x\neq2\)

\(x\neq-2,\,x\neq2\)

\(x\neq4\)

Explanation:

Solution: x²+4x+4=(x+2)²≠0.

Question 27

3.33 pts

Find the domain of \(f(x)=\dfrac{x+7}{x^2-49}\).

\(x\neq7,\,x\neq-7\)

\(x\neq49\)

\(x\neq7\)

\(x\neq-7\)

Explanation:

Solution: x²-49=(x-7)(x+7)≠0.

Question 28

3.33 pts

Find the domain of \(f(x)=\dfrac{2x^2-5}{x^3+8}\).

\(x\neq-2\)

\(x\neq8\)

\(x\neq2\)

\(x\neq-8\)

Explanation:

Solution: x³+8=(x+2)(x²-2x+4)≠0.

Question 29

3.33 pts

Find the domain of \(f(x)=\dfrac{x^2+10x+25}{x+5}\).

\(x\neq-5\)

\(x\neq5\)

\(x\neq-5,\,x\neq-2\)

\(x\neq-10\)

Explanation:

Solution: x+5≠0.

Question 30

3.33 pts

Find the domain of \(f(x)=\dfrac{x^3-1}{x^2-2x+1}\).

\(x\neq1\)

\(x\neq1,\,x\neq-1\)

\(x\neq2\)

\(x\neq-1\)

Explanation:

Solution: x²-2x+1=(x-1)²≠0.