Domain of Functions — Practice Questions

This page contains 20 practice questions on finding the domain of a function. The problems cover polynomial, rational, square root, logarithmic, exponential, absolute value, and set-of-ordered-pairs functions. Complete step-by-step solutions are provided directly beneath each question — ideal for high school and precalculus students.

Question 1 — Linear Polynomial

Find the domain of \( f(x) = 2x + 7 \).

View Solution

Polynomials have no division-by-zero or even-root restrictions, so they are defined for all real numbers.

Answer: \( (-\infty, \infty) \)

Question 2 — Rational Function

Find the domain of \( g(x) = \dfrac{1}{-x + 8} \).

View Solution

Denominator cannot be zero:

\[ -x + 8 = 0 \implies x = 8 \]

Answer: \( (-\infty, 8) \cup (8, \infty) \)

Question 3 — Rational Function

Find the domain of \( h(x) = \dfrac{x - 6}{2x - 8} \).

View Solution
\[ 2x - 8 \neq 0 \implies x \neq 4 \]

Answer: \( (-\infty, 4) \cup (4, \infty) \)

Question 4 — Square Root in the Denominator

Find the domain of \( f(x) = \dfrac{5}{\sqrt{-x + 4}} \).

View Solution

Radicand must be strictly positive (it's in the denominator):

\[ -x + 4 > 0 \implies x < 4 \]

Answer: \( (-\infty, 4) \)

Question 5 — Radical over Radical

Find the domain of \( h(x) = \dfrac{\sqrt{-x + 5}}{\sqrt{x - 3}} \).

View Solution
  1. Numerator radicand \(\ge 0\): \(x \le 5\)
  2. Denominator radicand \(> 0\): \(x > 3\)

Intersection: \( 3 < x \le 5 \).

Answer: \( (3, 5] \)

Question 6 — Absolute Value Under a Square Root

Find the domain of \( f(x) = \sqrt{|x| - 5} \).

View Solution
\[ |x| - 5 \ge 0 \implies |x| \ge 5 \implies x \ge 5 \text{ or } x \le -5 \]

Answer: \( (-\infty, -5] \cup [5, \infty) \)

Question 7 — Linear Inside Absolute Value

Find the domain of \( f(x) = \sqrt{|3x - 8| - 5} \).

View Solution
\[ |3x - 8| \ge 5 \]

Case 1: \(3x - 8 \ge 5 \implies x \ge \tfrac{13}{3}\)

Case 2: \(3x - 8 \le -5 \implies x \le 1\)

Answer: \( (-\infty, 1] \cup \left[\tfrac{13}{3}, \infty\right) \)

Question 8 — Rational with Quadratic Denominator

Find the domain of \( f(x) = \dfrac{3x - 6}{x^2 + 2x - 8} \).

View Solution
\[ x^2 + 2x - 8 = (x + 4)(x - 2) \neq 0 \]

So \(x \neq -4\) and \(x \neq 2\).

Answer: \( (-\infty, -4) \cup (-4, 2) \cup (2, \infty) \)

Question 9 — Quadratic Under a Denominator Radical

Find the domain of \( h(x) = \dfrac{3x - 6}{\sqrt{x^2 + 2x - 8}} \).

View Solution
\[ x^2 + 2x - 8 = (x + 4)(x - 2) > 0 \]

Upward parabola → positive outside the roots.

Answer: \( (-\infty, -4) \cup (2, \infty) \)

Question 10 — Absolute Value Function

Find the domain of \( g(x) = |2x - 4| \).

View Solution

Absolute value is defined for all reals.

Answer: \( (-\infty, \infty) \)

Question 11 — Natural Log of Absolute Value

Find the domain of \( h(x) = \ln(|2x - 9|) \).

View Solution
\[ |2x - 9| > 0 \implies 2x - 9 \neq 0 \implies x \neq \tfrac{9}{2} \]

Answer: \( \left(-\infty, \tfrac{9}{2}\right) \cup \left(\tfrac{9}{2}, \infty\right) \)

Question 12 — Rational with No Real Roots

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

View Solution

\(x^2 + 6 \ge 6 > 0\) for all real \(x\), so the denominator never vanishes.

Answer: \( (-\infty, \infty) \)

Question 13 — Quadratic with Negative Discriminant

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

View Solution
\[ \Delta = 2^2 - 4(1)(6) = -20 < 0 \]

Leading coefficient positive + no real roots → quadratic is always positive.

Answer: \( (-\infty, \infty) \)

Question 14 — Nested Absolute Value and Radical

Find the domain of \( h(x) = \dfrac{1}{\sqrt{|-x^2 - 9|}} \).

View Solution
\[ |-x^2 - 9| = |x^2 + 9| = x^2 + 9 \ge 9 > 0 \]

The radicand is always positive, so the denominator is never zero.

Answer: \( (-\infty, \infty) \)

Question 15 — Domain from Ordered Pairs

Find the domain of the relation \( g = \{(3,2), (5,9), (6,9), (11,0)\} \).

View Solution

The domain is the set of first coordinates.

Answer: \( \{3, 5, 6, 11\} \)

Question 16 — Radical in a Denominator with Shift

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

View Solution
  1. Radicand: \(x + 2 \ge 0 \implies x \ge -2\)
  2. Denominator: \(2 - \sqrt{x + 2} \neq 0 \implies \sqrt{x+2} \neq 2 \implies x \neq 2\)

Answer: \( [-2, 2) \cup (2, \infty) \)

Question 17 — Exponential of a Logarithm

Find the domain of \( f(x) = e^{\ln(x)} \).

View Solution

The inner \(\ln(x)\) requires \(x > 0\). The outer \(e^u\) is defined for all \(u\).

Answer: \( (0, \infty) \)

Question 18 — Logarithm in the Denominator

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

View Solution
  1. Log argument positive: \(2x - 3 > 0 \implies x > \tfrac{3}{2}\)
  2. Denominator nonzero: \(\log(2x - 3) \neq 0 \implies 2x - 3 \neq 1 \implies x \neq 2\)

Answer: \( \left(\tfrac{3}{2}, 2\right) \cup (2, \infty) \)

Question 19 — Natural Log of a Linear

Find the domain of \( g(x) = \ln(2x - 8) \).

View Solution
\[ 2x - 8 > 0 \implies x > 4 \]

Answer: \( (4, \infty) \)

Question 20 — Log of a Factorable Quadratic

Find the domain of \( g(x) = \log(x^2 + 3x - 10) \).

View Solution
\[ x^2 + 3x - 10 = (x + 5)(x - 2) > 0 \]

Upward parabola → positive outside the roots.

Answer: \( (-\infty, -5) \cup (2, \infty) \)

Frequently Asked Questions

What is the domain of a function?

The domain is the set of all real input values \(x\) for which the function produces a real output.

How do I find the domain of a rational function?

Set the denominator equal to zero and exclude those values from the real numbers. The domain is all reals except the zeros of the denominator.

What is the domain of a square root function?

Require the radicand to be nonnegative: solve \(\text{radicand} \ge 0\). If the radical is in a denominator, use \(> 0\) instead to avoid division by zero.

What is the domain of a logarithmic function?

Require the argument of the logarithm to be strictly positive. Also, if the log appears in a denominator, exclude where the log equals zero.

How do I write the domain in interval notation?

Use parentheses \( (a, b) \) for excluded endpoints, brackets \( [a, b] \) for included endpoints, and \(\cup\) to join separate intervals.