Domain of a Rational Function — Dynamic Practice

Domain of a Rational Function — Dynamic Practice. Practice questions to deepen understanding of the domain of rational (quotient) functions. Online dynamic learning math system.

Dynamic practice in the domain of rational functions — the denominator must be non-zero. New questions every attempt for unlimited practice.

42 questions

Question 1

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-441}\)

\(x \neq 21, x \neq -21\)

\(x \neq 21\)

\(x > 21,\ x < -21\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-441}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 441 \neq 0\)

📌 Step 2: Factor the expression

\((x - 21)(x + 21) \neq 0\)

📌 Step 3: Each factor separately

\(x - 21 \neq 0 \;\Rightarrow\; x \neq 21\)

\(x + 21 \neq 0 \;\Rightarrow\; x \neq -21\)

Domain: \(x \neq 21, x \neq -21\)

Question 2

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x-17}\)

\(x \neq 17\)

\(x \geq 17\)

\(x > 17\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x-17}\)

📌 Step 1: The denominator cannot be zero

\(x - 17 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 17\)

Domain: \(x \neq 17\)

Question 3

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x-20}\)

\(x \neq 20\)

\(x \geq 20\)

\(x > 20\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x-20}\)

📌 Step 1: The denominator cannot be zero

\(x - 20 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 20\)

Domain: \(x \neq 20\)

Question 4

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x-12}\)

\(x \neq 12\)

\(x \geq 12\)

\(x > 12\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x-12}\)

📌 Step 1: The denominator cannot be zero

\(x - 12 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 12\)

Domain: \(x \neq 12\)

Question 5

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+20}\)

\(x \neq -20\)

\(x \geq -20\)

\(x > -20\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+20}\)

📌 Step 1: The denominator cannot be zero

\(x + 20 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -20\)

Domain: \(x \neq -20\)

Question 6

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-121}\)

\(x \neq 11, x \neq -11\)

\(x \neq 11\)

\(x > 11,\ x < -11\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-121}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 121 \neq 0\)

📌 Step 2: Factor the expression

\((x - 11)(x + 11) \neq 0\)

📌 Step 3: Each factor separately

\(x - 11 \neq 0 \;\Rightarrow\; x \neq 11\)

\(x + 11 \neq 0 \;\Rightarrow\; x \neq -11\)

Domain: \(x \neq 11, x \neq -11\)

Question 7

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x-19}\)

\(x \neq 19\)

\(x \geq 19\)

\(x > 19\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x-19}\)

📌 Step 1: The denominator cannot be zero

\(x - 19 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 19\)

Domain: \(x \neq 19\)

Question 8

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-324}\)

\(x \neq 18, x \neq -18\)

\(x \neq 18\)

\(x > 18,\ x < -18\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-324}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 324 \neq 0\)

📌 Step 2: Factor the expression

\((x - 18)(x + 18) \neq 0\)

📌 Step 3: Each factor separately

\(x - 18 \neq 0 \;\Rightarrow\; x \neq 18\)

\(x + 18 \neq 0 \;\Rightarrow\; x \neq -18\)

Domain: \(x \neq 18, x \neq -18\)

Question 9

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-16}\)

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

\(x \neq 4\)

\(x > 4,\ x < -4\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-16}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 16 \neq 0\)

📌 Step 2: Factor the expression

\((x - 4)(x + 4) \neq 0\)

📌 Step 3: Each factor separately

\(x - 4 \neq 0 \;\Rightarrow\; x \neq 4\)

\(x + 4 \neq 0 \;\Rightarrow\; x \neq -4\)

Domain: \(x \neq 4, x \neq -4\)

Question 10

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+15}\)

\(x \neq -15\)

\(x \geq -15\)

\(x > -15\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+15}\)

📌 Step 1: The denominator cannot be zero

\(x + 15 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -15\)

Domain: \(x \neq -15\)

Question 11

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+21}\)

\(x \neq -21\)

\(x \geq -21\)

\(x > -21\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+21}\)

📌 Step 1: The denominator cannot be zero

\(x + 21 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -21\)

Domain: \(x \neq -21\)

Question 12

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x+11}\)

\(x \neq -11\)

\(x \geq -11\)

\(x > -11\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x+11}\)

📌 Step 1: The denominator cannot be zero

\(x + 11 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -11\)

Domain: \(x \neq -11\)

Question 13

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x+9}\)

\(x \neq -9\)

\(x \geq -9\)

\(x > -9\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x+9}\)

📌 Step 1: The denominator cannot be zero

\(x + 9 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -9\)

Domain: \(x \neq -9\)

Question 14

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x-10}\)

\(x \neq 10\)

\(x \geq 10\)

\(x > 10\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x-10}\)

📌 Step 1: The denominator cannot be zero

\(x - 10 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 10\)

Domain: \(x \neq 10\)

Question 15

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x-8}\)

\(x \neq 8\)

\(x \geq 8\)

\(x > 8\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x-8}\)

📌 Step 1: The denominator cannot be zero

\(x - 8 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 8\)

Domain: \(x \neq 8\)

Question 16

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x-24}\)

\(x \neq 24\)

\(x \geq 24\)

\(x > 24\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x-24}\)

📌 Step 1: The denominator cannot be zero

\(x - 24 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 24\)

Domain: \(x \neq 24\)

Question 17

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-49}\)

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

\(x \neq 7\)

\(x > 7,\ x < -7\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-49}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 49 \neq 0\)

📌 Step 2: Factor the expression

\((x - 7)(x + 7) \neq 0\)

📌 Step 3: Each factor separately

\(x - 7 \neq 0 \;\Rightarrow\; x \neq 7\)

\(x + 7 \neq 0 \;\Rightarrow\; x \neq -7\)

Domain: \(x \neq 7, x \neq -7\)

Question 18

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+14}\)

\(x \neq -14\)

\(x \geq -14\)

\(x > -14\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+14}\)

📌 Step 1: The denominator cannot be zero

\(x + 14 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -14\)

Domain: \(x \neq -14\)

Question 19

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-289}\)

\(x \neq 17, x \neq -17\)

\(x \neq 17\)

\(x > 17,\ x < -17\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-289}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 289 \neq 0\)

📌 Step 2: Factor the expression

\((x - 17)(x + 17) \neq 0\)

📌 Step 3: Each factor separately

\(x - 17 \neq 0 \;\Rightarrow\; x \neq 17\)

\(x + 17 \neq 0 \;\Rightarrow\; x \neq -17\)

Domain: \(x \neq 17, x \neq -17\)

Question 20

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x+14}\)

\(x \neq -14\)

\(x \geq -14\)

\(x > -14\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x+14}\)

📌 Step 1: The denominator cannot be zero

\(x + 14 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -14\)

Domain: \(x \neq -14\)

Question 21

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x-21}\)

\(x \neq 21\)

\(x \geq 21\)

\(x > 21\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x-21}\)

📌 Step 1: The denominator cannot be zero

\(x - 21 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 21\)

Domain: \(x \neq 21\)

Question 22

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x-19}\)

\(x \neq 19\)

\(x \geq 19\)

\(x > 19\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x-19}\)

📌 Step 1: The denominator cannot be zero

\(x - 19 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 19\)

Domain: \(x \neq 19\)

Question 23

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x+23}\)

\(x \neq -23\)

\(x \geq -23\)

\(x > -23\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x+23}\)

📌 Step 1: The denominator cannot be zero

\(x + 23 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -23\)

Domain: \(x \neq -23\)

Question 24

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x-2}\)

\(x \neq 2\)

\(x \geq 2\)

\(x > 2\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x-2}\)

📌 Step 1: The denominator cannot be zero

\(x - 2 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 2\)

Domain: \(x \neq 2\)

Question 25

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x-6}\)

\(x \neq 6\)

\(x \geq 6\)

\(x > 6\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x-6}\)

📌 Step 1: The denominator cannot be zero

\(x - 6 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 6\)

Domain: \(x \neq 6\)

Question 26

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+10}\)

\(x \neq -10\)

\(x \geq -10\)

\(x > -10\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+10}\)

📌 Step 1: The denominator cannot be zero

\(x + 10 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -10\)

Domain: \(x \neq -10\)

Question 27

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x-5}\)

\(x \neq 5\)

\(x \geq 5\)

\(x > 5\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x-5}\)

📌 Step 1: The denominator cannot be zero

\(x - 5 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 5\)

Domain: \(x \neq 5\)

Question 28

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+24}\)

\(x \neq -24\)

\(x \geq -24\)

\(x > -24\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+24}\)

📌 Step 1: The denominator cannot be zero

\(x + 24 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -24\)

Domain: \(x \neq -24\)

Question 29

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+4}\)

\(x \neq -4\)

\(x \geq -4\)

\(x > -4\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+4}\)

📌 Step 1: The denominator cannot be zero

\(x + 4 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -4\)

Domain: \(x \neq -4\)

Question 30

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-529}\)

\(x \neq 23, x \neq -23\)

\(x \neq 23\)

\(x > 23,\ x < -23\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-529}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 529 \neq 0\)

📌 Step 2: Factor the expression

\((x - 23)(x + 23) \neq 0\)

📌 Step 3: Each factor separately

\(x - 23 \neq 0 \;\Rightarrow\; x \neq 23\)

\(x + 23 \neq 0 \;\Rightarrow\; x \neq -23\)

Domain: \(x \neq 23, x \neq -23\)

Question 31

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+1}\)

\(x \neq -1\)

\(x \geq -1\)

\(x > -1\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+1}\)

📌 Step 1: The denominator cannot be zero

\(x + 1 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -1\)

Domain: \(x \neq -1\)

Question 32

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-81}\)

\(x \neq 9, x \neq -9\)

\(x \neq 9\)

\(x > 9,\ x < -9\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-81}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 81 \neq 0\)

📌 Step 2: Factor the expression

\((x - 9)(x + 9) \neq 0\)

📌 Step 3: Each factor separately

\(x - 9 \neq 0 \;\Rightarrow\; x \neq 9\)

\(x + 9 \neq 0 \;\Rightarrow\; x \neq -9\)

Domain: \(x \neq 9, x \neq -9\)

Question 33

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-25}\)

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

\(x \neq 5\)

\(x > 5,\ x < -5\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-25}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 25 \neq 0\)

📌 Step 2: Factor the expression

\((x - 5)(x + 5) \neq 0\)

📌 Step 3: Each factor separately

\(x - 5 \neq 0 \;\Rightarrow\; x \neq 5\)

\(x + 5 \neq 0 \;\Rightarrow\; x \neq -5\)

Domain: \(x \neq 5, x \neq -5\)

Question 34

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-625}\)

\(x \neq 25, x \neq -25\)

\(x \neq 25\)

\(x > 25,\ x < -25\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-625}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 625 \neq 0\)

📌 Step 2: Factor the expression

\((x - 25)(x + 25) \neq 0\)

📌 Step 3: Each factor separately

\(x - 25 \neq 0 \;\Rightarrow\; x \neq 25\)

\(x + 25 \neq 0 \;\Rightarrow\; x \neq -25\)

Domain: \(x \neq 25, x \neq -25\)

Question 35

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-361}\)

\(x \neq 19, x \neq -19\)

\(x \neq 19\)

\(x > 19,\ x < -19\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-361}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 361 \neq 0\)

📌 Step 2: Factor the expression

\((x - 19)(x + 19) \neq 0\)

📌 Step 3: Each factor separately

\(x - 19 \neq 0 \;\Rightarrow\; x \neq 19\)

\(x + 19 \neq 0 \;\Rightarrow\; x \neq -19\)

Domain: \(x \neq 19, x \neq -19\)

Question 36

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x-7}\)

\(x \neq 7\)

\(x \geq 7\)

\(x > 7\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x-7}\)

📌 Step 1: The denominator cannot be zero

\(x - 7 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 7\)

Domain: \(x \neq 7\)

Question 37

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x^2-400}\)

\(x \neq 20, x \neq -20\)

\(x \neq 20\)

\(x > 20,\ x < -20\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x^2-400}\)

📌 Step 1: The denominator cannot be zero

\(x^2 - 400 \neq 0\)

📌 Step 2: Factor the expression

\((x - 20)(x + 20) \neq 0\)

📌 Step 3: Each factor separately

\(x - 20 \neq 0 \;\Rightarrow\; x \neq 20\)

\(x + 20 \neq 0 \;\Rightarrow\; x \neq -20\)

Domain: \(x \neq 20, x \neq -20\)

Question 38

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x+1}\)

\(x \neq -1\)

\(x \geq -1\)

\(x > -1\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x+1}\)

📌 Step 1: The denominator cannot be zero

\(x + 1 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -1\)

Domain: \(x \neq -1\)

Question 39

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+9}\)

\(x \neq -9\)

\(x \geq -9\)

\(x > -9\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+9}\)

📌 Step 1: The denominator cannot be zero

\(x + 9 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -9\)

Domain: \(x \neq -9\)

Question 40

2.38 pts

Find the domain of the function:

\(f(x) = \frac{x}{x+12}\)

\(x \neq -12\)

\(x \geq -12\)

\(x > -12\)

\(x \neq 0\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{x}{x+12}\)

📌 Step 1: The denominator cannot be zero

\(x + 12 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -12\)

Domain: \(x \neq -12\)

Question 41

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x+8}\)

\(x \neq -8\)

\(x \geq -8\)

\(x > -8\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x+8}\)

📌 Step 1: The denominator cannot be zero

\(x + 8 \neq 0\)

📌 Step 2: Isolate x

\(x \neq -8\)

Domain: \(x \neq -8\)

Question 42

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{x-3}\)

\(x \neq 3\)

\(x \geq 3\)

\(x > 3\)

\(\mathbb{R}\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{x-3}\)

📌 Step 1: The denominator cannot be zero

\(x - 3 \neq 0\)

📌 Step 2: Isolate x

\(x \neq 3\)

Domain: \(x \neq 3\)