Domain — Polynomial Square Root in the Denominator — Dynamic Practice

Domain — Polynomial Square Root in the Denominator — Dynamic Practice. Practice questions to deepen understanding of the domain when a polynomial square root appears in the denominator. Interactive math practice with instant feedback.

Dynamic practice in the domain when √(polynomial) appears in the denominator — the polynomial must be strictly positive (not just non-negative). New questions every attempt.

42 questions

Question 1

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+3)(x-4)}}\)

\(x < -3 \text{ or } x > 4\)

\(-3 < x < 4\)

\(x \leq -3 \text{ or } x \geq 4\)

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

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+3)(x-4)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -3)(x - 4) > 0\)

📌 Step 2: Roots of the expression

\(x = -3\) or \(x = 4\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -3	-3	-3 < x < 4	4	x > 4
Sign	+	0	−	0	+

Strict positivity required: \(x < -3\) or \(x > 4\)

Domain: \(x < -3 \text{ or } x > 4\)

Question 2

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+12)(x+8)}}\)

\(x < -12 \text{ or } x > -8\)

\(-12 < x < -8\)

\(x \leq -12 \text{ or } x \geq -8\)

\(x \neq -12, x \neq -8\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+12)(x+8)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -12)(x - -8) > 0\)

📌 Step 2: Roots of the expression

\(x = -12\) or \(x = -8\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -12	-12	-12 < x < -8	-8	x > -8
Sign	+	0	−	0	+

Strict positivity required: \(x < -12\) or \(x > -8\)

Domain: \(x < -12 \text{ or } x > -8\)

Question 3

2.38 pts

Find the domain of the function:

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

\(x < 1 \text{ or } x > 3\)

\(1 < x < 3\)

\(x \leq 1 \text{ or } x \geq 3\)

\(x \neq 1, x \neq 3\)

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 1)(x - 3) > 0\)

📌 Step 2: Roots of the expression

\(x = 1\) or \(x = 3\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 1	1	1 < x < 3	3	x > 3
Sign	+	0	−	0	+

Strict positivity required: \(x < 1\) or \(x > 3\)

Domain: \(x < 1 \text{ or } x > 3\)

Question 4

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-1)(x-4)}}\)

\(x < 1 \text{ or } x > 4\)

\(1 < x < 4\)

\(x \leq 1 \text{ or } x \geq 4\)

\(x \neq 1, x \neq 4\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-1)(x-4)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 1)(x - 4) > 0\)

📌 Step 2: Roots of the expression

\(x = 1\) or \(x = 4\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 1	1	1 < x < 4	4	x > 4
Sign	+	0	−	0	+

Strict positivity required: \(x < 1\) or \(x > 4\)

Domain: \(x < 1 \text{ or } x > 4\)

Question 5

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+15)(x+13)}}\)

\(x < -15 \text{ or } x > -13\)

\(-15 < x < -13\)

\(x \leq -15 \text{ or } x \geq -13\)

\(x \neq -15, x \neq -13\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+15)(x+13)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -15)(x - -13) > 0\)

📌 Step 2: Roots of the expression

\(x = -15\) or \(x = -13\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -15	-15	-15 < x < -13	-13	x > -13
Sign	+	0	−	0	+

Strict positivity required: \(x < -15\) or \(x > -13\)

Domain: \(x < -15 \text{ or } x > -13\)

Question 6

2.38 pts

Find the domain of the function:

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

\(x < -2 \text{ or } x > 1\)

\(-2 < x < 1\)

\(x \leq -2 \text{ or } x \geq 1\)

\(x \neq -2, x \neq 1\)

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -2)(x - 1) > 0\)

📌 Step 2: Roots of the expression

\(x = -2\) or \(x = 1\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -2	-2	-2 < x < 1	1	x > 1
Sign	+	0	−	0	+

Strict positivity required: \(x < -2\) or \(x > 1\)

Domain: \(x < -2 \text{ or } x > 1\)

Question 7

2.38 pts

Find the domain of the function:

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

\(x < 0 \text{ or } x > 7\)

\(0 < x < 7\)

\(x \leq 0 \text{ or } x \geq 7\)

\(x \neq 0, x \neq 7\)

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 0)(x - 7) > 0\)

📌 Step 2: Roots of the expression

\(x = 0\) or \(x = 7\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 0	0	0 < x < 7	7	x > 7
Sign	+	0	−	0	+

Strict positivity required: \(x < 0\) or \(x > 7\)

Domain: \(x < 0 \text{ or } x > 7\)

Question 8

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-3)(x-8)}}\)

\(x < 3 \text{ or } x > 8\)

\(3 < x < 8\)

\(x \leq 3 \text{ or } x \geq 8\)

\(x \neq 3, x \neq 8\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-3)(x-8)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 3)(x - 8) > 0\)

📌 Step 2: Roots of the expression

\(x = 3\) or \(x = 8\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 3	3	3 < x < 8	8	x > 8
Sign	+	0	−	0	+

Strict positivity required: \(x < 3\) or \(x > 8\)

Domain: \(x < 3 \text{ or } x > 8\)

Question 9

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-2)(x-9)}}\)

\(x < 2 \text{ or } x > 9\)

\(2 < x < 9\)

\(x \leq 2 \text{ or } x \geq 9\)

\(x \neq 2, x \neq 9\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-2)(x-9)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 2)(x - 9) > 0\)

📌 Step 2: Roots of the expression

\(x = 2\) or \(x = 9\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 2	2	2 < x < 9	9	x > 9
Sign	+	0	−	0	+

Strict positivity required: \(x < 2\) or \(x > 9\)

Domain: \(x < 2 \text{ or } x > 9\)

Question 10

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+9)(x+6)}}\)

\(x < -9 \text{ or } x > -6\)

\(-9 < x < -6\)

\(x \leq -9 \text{ or } x \geq -6\)

\(x \neq -9, x \neq -6\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+9)(x+6)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -9)(x - -6) > 0\)

📌 Step 2: Roots of the expression

\(x = -9\) or \(x = -6\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -9	-9	-9 < x < -6	-6	x > -6
Sign	+	0	−	0	+

Strict positivity required: \(x < -9\) or \(x > -6\)

Domain: \(x < -9 \text{ or } x > -6\)

Question 11

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+9)(x+7)}}\)

\(x < -9 \text{ or } x > -7\)

\(-9 < x < -7\)

\(x \leq -9 \text{ or } x \geq -7\)

\(x \neq -9, x \neq -7\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+9)(x+7)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -9)(x - -7) > 0\)

📌 Step 2: Roots of the expression

\(x = -9\) or \(x = -7\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -9	-9	-9 < x < -7	-7	x > -7
Sign	+	0	−	0	+

Strict positivity required: \(x < -9\) or \(x > -7\)

Domain: \(x < -9 \text{ or } x > -7\)

Question 12

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+7)(x)}}\)

\(x < -7 \text{ or } x > 0\)

\(-7 < x < 0\)

\(x \leq -7 \text{ or } x \geq 0\)

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

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -7)(x - 0) > 0\)

📌 Step 2: Roots of the expression

\(x = -7\) or \(x = 0\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -7	-7	-7 < x < 0	0	x > 0
Sign	+	0	−	0	+

Strict positivity required: \(x < -7\) or \(x > 0\)

Domain: \(x < -7 \text{ or } x > 0\)

Question 13

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+15)(x+9)}}\)

\(x < -15 \text{ or } x > -9\)

\(-15 < x < -9\)

\(x \leq -15 \text{ or } x \geq -9\)

\(x \neq -15, x \neq -9\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+15)(x+9)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -15)(x - -9) > 0\)

📌 Step 2: Roots of the expression

\(x = -15\) or \(x = -9\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -15	-15	-15 < x < -9	-9	x > -9
Sign	+	0	−	0	+

Strict positivity required: \(x < -15\) or \(x > -9\)

Domain: \(x < -15 \text{ or } x > -9\)

Question 14

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-4)(x-6)}}\)

\(x < 4 \text{ or } x > 6\)

\(4 < x < 6\)

\(x \leq 4 \text{ or } x \geq 6\)

\(x \neq 4, x \neq 6\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-4)(x-6)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 4)(x - 6) > 0\)

📌 Step 2: Roots of the expression

\(x = 4\) or \(x = 6\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 4	4	4 < x < 6	6	x > 6
Sign	+	0	−	0	+

Strict positivity required: \(x < 4\) or \(x > 6\)

Domain: \(x < 4 \text{ or } x > 6\)

Question 15

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+6)(x-2)}}\)

\(x < -6 \text{ or } x > 2\)

\(-6 < x < 2\)

\(x \leq -6 \text{ or } x \geq 2\)

\(x \neq -6, x \neq 2\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+6)(x-2)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -6)(x - 2) > 0\)

📌 Step 2: Roots of the expression

\(x = -6\) or \(x = 2\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -6	-6	-6 < x < 2	2	x > 2
Sign	+	0	−	0	+

Strict positivity required: \(x < -6\) or \(x > 2\)

Domain: \(x < -6 \text{ or } x > 2\)

Question 16

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-5)(x-13)}}\)

\(x < 5 \text{ or } x > 13\)

\(5 < x < 13\)

\(x \leq 5 \text{ or } x \geq 13\)

\(x \neq 5, x \neq 13\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-5)(x-13)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 5)(x - 13) > 0\)

📌 Step 2: Roots of the expression

\(x = 5\) or \(x = 13\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 5	5	5 < x < 13	13	x > 13
Sign	+	0	−	0	+

Strict positivity required: \(x < 5\) or \(x > 13\)

Domain: \(x < 5 \text{ or } x > 13\)

Question 17

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+11)(x+8)}}\)

\(x < -11 \text{ or } x > -8\)

\(-11 < x < -8\)

\(x \leq -11 \text{ or } x \geq -8\)

\(x \neq -11, x \neq -8\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+11)(x+8)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -11)(x - -8) > 0\)

📌 Step 2: Roots of the expression

\(x = -11\) or \(x = -8\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -11	-11	-11 < x < -8	-8	x > -8
Sign	+	0	−	0	+

Strict positivity required: \(x < -11\) or \(x > -8\)

Domain: \(x < -11 \text{ or } x > -8\)

Question 18

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+4)(x-2)}}\)

\(x < -4 \text{ or } x > 2\)

\(-4 < x < 2\)

\(x \leq -4 \text{ or } x \geq 2\)

\(x \neq -4, x \neq 2\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+4)(x-2)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -4)(x - 2) > 0\)

📌 Step 2: Roots of the expression

\(x = -4\) or \(x = 2\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -4	-4	-4 < x < 2	2	x > 2
Sign	+	0	−	0	+

Strict positivity required: \(x < -4\) or \(x > 2\)

Domain: \(x < -4 \text{ or } x > 2\)

Question 19

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+6)(x+2)}}\)

\(x < -6 \text{ or } x > -2\)

\(-6 < x < -2\)

\(x \leq -6 \text{ or } x \geq -2\)

\(x \neq -6, x \neq -2\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+6)(x+2)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -6)(x - -2) > 0\)

📌 Step 2: Roots of the expression

\(x = -6\) or \(x = -2\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -6	-6	-6 < x < -2	-2	x > -2
Sign	+	0	−	0	+

Strict positivity required: \(x < -6\) or \(x > -2\)

Domain: \(x < -6 \text{ or } x > -2\)

Question 20

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+5)(x)}}\)

\(x < -5 \text{ or } x > 0\)

\(-5 < x < 0\)

\(x \leq -5 \text{ or } x \geq 0\)

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

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+5)(x)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -5)(x - 0) > 0\)

📌 Step 2: Roots of the expression

\(x = -5\) or \(x = 0\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -5	-5	-5 < x < 0	0	x > 0
Sign	+	0	−	0	+

Strict positivity required: \(x < -5\) or \(x > 0\)

Domain: \(x < -5 \text{ or } x > 0\)

Question 21

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-3)(x-5)}}\)

\(x < 3 \text{ or } x > 5\)

\(3 < x < 5\)

\(x \leq 3 \text{ or } x \geq 5\)

\(x \neq 3, x \neq 5\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-3)(x-5)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 3)(x - 5) > 0\)

📌 Step 2: Roots of the expression

\(x = 3\) or \(x = 5\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 3	3	3 < x < 5	5	x > 5
Sign	+	0	−	0	+

Strict positivity required: \(x < 3\) or \(x > 5\)

Domain: \(x < 3 \text{ or } x > 5\)

Question 22

2.38 pts

Find the domain of the function:

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

\(x < -3 \text{ or } x > 1\)

\(-3 < x < 1\)

\(x \leq -3 \text{ or } x \geq 1\)

\(x \neq -3, x \neq 1\)

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -3)(x - 1) > 0\)

📌 Step 2: Roots of the expression

\(x = -3\) or \(x = 1\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -3	-3	-3 < x < 1	1	x > 1
Sign	+	0	−	0	+

Strict positivity required: \(x < -3\) or \(x > 1\)

Domain: \(x < -3 \text{ or } x > 1\)

Question 23

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-5)(x-12)}}\)

\(x < 5 \text{ or } x > 12\)

\(5 < x < 12\)

\(x \leq 5 \text{ or } x \geq 12\)

\(x \neq 5, x \neq 12\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-5)(x-12)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 5)(x - 12) > 0\)

📌 Step 2: Roots of the expression

\(x = 5\) or \(x = 12\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 5	5	5 < x < 12	12	x > 12
Sign	+	0	−	0	+

Strict positivity required: \(x < 5\) or \(x > 12\)

Domain: \(x < 5 \text{ or } x > 12\)

Question 24

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-3)(x-6)}}\)

\(x < 3 \text{ or } x > 6\)

\(3 < x < 6\)

\(x \leq 3 \text{ or } x \geq 6\)

\(x \neq 3, x \neq 6\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-3)(x-6)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 3)(x - 6) > 0\)

📌 Step 2: Roots of the expression

\(x = 3\) or \(x = 6\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 3	3	3 < x < 6	6	x > 6
Sign	+	0	−	0	+

Strict positivity required: \(x < 3\) or \(x > 6\)

Domain: \(x < 3 \text{ or } x > 6\)

Question 25

2.38 pts

Find the domain of the function:

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

\(x < 0 \text{ or } x > 2\)

\(0 < x < 2\)

\(x \leq 0 \text{ or } x \geq 2\)

\(x \neq 0, x \neq 2\)

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 0)(x - 2) > 0\)

📌 Step 2: Roots of the expression

\(x = 0\) or \(x = 2\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 0	0	0 < x < 2	2	x > 2
Sign	+	0	−	0	+

Strict positivity required: \(x < 0\) or \(x > 2\)

Domain: \(x < 0 \text{ or } x > 2\)

Question 26

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+5)(x-3)}}\)

\(x < -5 \text{ or } x > 3\)

\(-5 < x < 3\)

\(x \leq -5 \text{ or } x \geq 3\)

\(x \neq -5, x \neq 3\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+5)(x-3)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -5)(x - 3) > 0\)

📌 Step 2: Roots of the expression

\(x = -5\) or \(x = 3\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -5	-5	-5 < x < 3	3	x > 3
Sign	+	0	−	0	+

Strict positivity required: \(x < -5\) or \(x > 3\)

Domain: \(x < -5 \text{ or } x > 3\)

Question 27

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+2)(x-4)}}\)

\(x < -2 \text{ or } x > 4\)

\(-2 < x < 4\)

\(x \leq -2 \text{ or } x \geq 4\)

\(x \neq -2, x \neq 4\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+2)(x-4)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -2)(x - 4) > 0\)

📌 Step 2: Roots of the expression

\(x = -2\) or \(x = 4\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -2	-2	-2 < x < 4	4	x > 4
Sign	+	0	−	0	+

Strict positivity required: \(x < -2\) or \(x > 4\)

Domain: \(x < -2 \text{ or } x > 4\)

Question 28

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+2)(x-3)}}\)

\(x < -2 \text{ or } x > 3\)

\(-2 < x < 3\)

\(x \leq -2 \text{ or } x \geq 3\)

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

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+2)(x-3)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -2)(x - 3) > 0\)

📌 Step 2: Roots of the expression

\(x = -2\) or \(x = 3\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -2	-2	-2 < x < 3	3	x > 3
Sign	+	0	−	0	+

Strict positivity required: \(x < -2\) or \(x > 3\)

Domain: \(x < -2 \text{ or } x > 3\)

Question 29

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+14)(x+11)}}\)

\(x < -14 \text{ or } x > -11\)

\(-14 < x < -11\)

\(x \leq -14 \text{ or } x \geq -11\)

\(x \neq -14, x \neq -11\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+14)(x+11)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -14)(x - -11) > 0\)

📌 Step 2: Roots of the expression

\(x = -14\) or \(x = -11\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -14	-14	-14 < x < -11	-11	x > -11
Sign	+	0	−	0	+

Strict positivity required: \(x < -14\) or \(x > -11\)

Domain: \(x < -14 \text{ or } x > -11\)

Question 30

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+10)(x+3)}}\)

\(x < -10 \text{ or } x > -3\)

\(-10 < x < -3\)

\(x \leq -10 \text{ or } x \geq -3\)

\(x \neq -10, x \neq -3\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+10)(x+3)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -10)(x - -3) > 0\)

📌 Step 2: Roots of the expression

\(x = -10\) or \(x = -3\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -10	-10	-10 < x < -3	-3	x > -3
Sign	+	0	−	0	+

Strict positivity required: \(x < -10\) or \(x > -3\)

Domain: \(x < -10 \text{ or } x > -3\)

Question 31

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+11)(x+7)}}\)

\(x < -11 \text{ or } x > -7\)

\(-11 < x < -7\)

\(x \leq -11 \text{ or } x \geq -7\)

\(x \neq -11, x \neq -7\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+11)(x+7)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -11)(x - -7) > 0\)

📌 Step 2: Roots of the expression

\(x = -11\) or \(x = -7\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -11	-11	-11 < x < -7	-7	x > -7
Sign	+	0	−	0	+

Strict positivity required: \(x < -11\) or \(x > -7\)

Domain: \(x < -11 \text{ or } x > -7\)

Question 32

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-7)(x-14)}}\)

\(x < 7 \text{ or } x > 14\)

\(7 < x < 14\)

\(x \leq 7 \text{ or } x \geq 14\)

\(x \neq 7, x \neq 14\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-7)(x-14)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 7)(x - 14) > 0\)

📌 Step 2: Roots of the expression

\(x = 7\) or \(x = 14\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 7	7	7 < x < 14	14	x > 14
Sign	+	0	−	0	+

Strict positivity required: \(x < 7\) or \(x > 14\)

Domain: \(x < 7 \text{ or } x > 14\)

Question 33

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-2)(x-5)}}\)

\(x < 2 \text{ or } x > 5\)

\(2 < x < 5\)

\(x \leq 2 \text{ or } x \geq 5\)

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

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-2)(x-5)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 2)(x - 5) > 0\)

📌 Step 2: Roots of the expression

\(x = 2\) or \(x = 5\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 2	2	2 < x < 5	5	x > 5
Sign	+	0	−	0	+

Strict positivity required: \(x < 2\) or \(x > 5\)

Domain: \(x < 2 \text{ or } x > 5\)

Question 34

2.38 pts

Find the domain of the function:

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

\(x < 0 \text{ or } x > 6\)

\(0 < x < 6\)

\(x \leq 0 \text{ or } x \geq 6\)

\(x \neq 0, x \neq 6\)

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 0)(x - 6) > 0\)

📌 Step 2: Roots of the expression

\(x = 0\) or \(x = 6\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 0	0	0 < x < 6	6	x > 6
Sign	+	0	−	0	+

Strict positivity required: \(x < 0\) or \(x > 6\)

Domain: \(x < 0 \text{ or } x > 6\)

Question 35

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+10)(x+5)}}\)

\(x < -10 \text{ or } x > -5\)

\(-10 < x < -5\)

\(x \leq -10 \text{ or } x \geq -5\)

\(x \neq -10, x \neq -5\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+10)(x+5)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -10)(x - -5) > 0\)

📌 Step 2: Roots of the expression

\(x = -10\) or \(x = -5\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -10	-10	-10 < x < -5	-5	x > -5
Sign	+	0	−	0	+

Strict positivity required: \(x < -10\) or \(x > -5\)

Domain: \(x < -10 \text{ or } x > -5\)

Question 36

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-2)(x-7)}}\)

\(x < 2 \text{ or } x > 7\)

\(2 < x < 7\)

\(x \leq 2 \text{ or } x \geq 7\)

\(x \neq 2, x \neq 7\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x-2)(x-7)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 2)(x - 7) > 0\)

📌 Step 2: Roots of the expression

\(x = 2\) or \(x = 7\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 2	2	2 < x < 7	7	x > 7
Sign	+	0	−	0	+

Strict positivity required: \(x < 2\) or \(x > 7\)

Domain: \(x < 2 \text{ or } x > 7\)

Question 37

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+12)(x+6)}}\)

\(x < -12 \text{ or } x > -6\)

\(-12 < x < -6\)

\(x \leq -12 \text{ or } x \geq -6\)

\(x \neq -12, x \neq -6\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+12)(x+6)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -12)(x - -6) > 0\)

📌 Step 2: Roots of the expression

\(x = -12\) or \(x = -6\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -12	-12	-12 < x < -6	-6	x > -6
Sign	+	0	−	0	+

Strict positivity required: \(x < -12\) or \(x > -6\)

Domain: \(x < -12 \text{ or } x > -6\)

Question 38

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+14)(x+8)}}\)

\(x < -14 \text{ or } x > -8\)

\(-14 < x < -8\)

\(x \leq -14 \text{ or } x \geq -8\)

\(x \neq -14, x \neq -8\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+14)(x+8)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -14)(x - -8) > 0\)

📌 Step 2: Roots of the expression

\(x = -14\) or \(x = -8\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -14	-14	-14 < x < -8	-8	x > -8
Sign	+	0	−	0	+

Strict positivity required: \(x < -14\) or \(x > -8\)

Domain: \(x < -14 \text{ or } x > -8\)

Question 39

2.38 pts

Find the domain of the function:

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

\(x < -1 \text{ or } x > 2\)

\(-1 < x < 2\)

\(x \leq -1 \text{ or } x \geq 2\)

\(x \neq -1, x \neq 2\)

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -1)(x - 2) > 0\)

📌 Step 2: Roots of the expression

\(x = -1\) or \(x = 2\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -1	-1	-1 < x < 2	2	x > 2
Sign	+	0	−	0	+

Strict positivity required: \(x < -1\) or \(x > 2\)

Domain: \(x < -1 \text{ or } x > 2\)

Question 40

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+8)(x+2)}}\)

\(x < -8 \text{ or } x > -2\)

\(-8 < x < -2\)

\(x \leq -8 \text{ or } x \geq -2\)

\(x \neq -8, x \neq -2\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+8)(x+2)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -8)(x - -2) > 0\)

📌 Step 2: Roots of the expression

\(x = -8\) or \(x = -2\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -8	-8	-8 < x < -2	-2	x > -2
Sign	+	0	−	0	+

Strict positivity required: \(x < -8\) or \(x > -2\)

Domain: \(x < -8 \text{ or } x > -2\)

Question 41

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x-1)(x-9)}}\)

\(x < 1 \text{ or } x > 9\)

\(1 < x < 9\)

\(x \leq 1 \text{ or } x \geq 9\)

\(x \neq 1, x \neq 9\)

Explanation:

Solution — Domain:

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

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - 1)(x - 9) > 0\)

📌 Step 2: Roots of the expression

\(x = 1\) or \(x = 9\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < 1	1	1 < x < 9	9	x > 9
Sign	+	0	−	0	+

Strict positivity required: \(x < 1\) or \(x > 9\)

Domain: \(x < 1 \text{ or } x > 9\)

Question 42

2.38 pts

Find the domain of the function:

\(f(x) = \frac{1}{\sqrt{(x+4)(x+2)}}\)

\(x < -4 \text{ or } x > -2\)

\(-4 < x < -2\)

\(x \leq -4 \text{ or } x \geq -2\)

\(x \neq -4, x \neq -2\)

Explanation:

Solution — Domain:

The function: \(f(x) = \frac{1}{\sqrt{(x+4)(x+2)}}\)

📌 Step 1: Square root in denominator — the product must be strictly positive

\((x - -4)(x - -2) > 0\)

📌 Step 2: Roots of the expression

\(x = -4\) or \(x = -2\)

📌 Step 3: Sign table (upward-opening parabola — positive outside the roots)

x	x < -4	-4	-4 < x < -2	-2	x > -2
Sign	+	0	−	0	+

Strict positivity required: \(x < -4\) or \(x > -2\)

Domain: \(x < -4 \text{ or } x > -2\)