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Domain and Range

Find the domain and range of various functions expressed in different ways
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This is level 2; Describe the domain in set notation. You can earn a trophy if you get at least 7 questions correct.

1. What is the domain of \(f(x) = \dfrac{1}{x+4}\)?

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2. What is the domain of \(g(x) = \dfrac{8}{x-5}\)?

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3. What is the domain of \(h(x) = \dfrac{4}{(x-3)(x+6)}\)?

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4. What is the domain of \(f(x) = \dfrac{3x}{x^2-36}\)?

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5. What is the domain of \(r(y) = \sqrt{y-6}\)?

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6. What is the domain of \(p(x) = \sqrt{x+5}\)?

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7. What is the domain of \(f(a) = \dfrac{1}{\sqrt{a-2}}\)?

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8. What is the domain of \(g(x) = \dfrac{2x}{\sqrt{5-2x}}\)?

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9. What is the domain of \(a(x) = \dfrac{3}{x} + \dfrac{2}{x-4}\)?

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10. What is the domain of \(f(x) = \sqrt{x^2-1}\)?

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This is Domain and Range level 2. You can also try:
Level 1 Level 3 Level 4

Instructions

Try your best to answer the questions above. Choose one of the five possible answers. When you have finished click the "check" button. If you have any questions wrong, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

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Description of Levels

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Domain, Range and Asymptotes - A simple matching puzzle to get you started.

Level 1 - Describe the domain in plain English

Level 2 - Describe the domain in set notation

Level 3 - Describe the domain using interval notation

Level 4 - Describe the domain and range using a mix of notations

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Understanding Domain and Range Notation

Level 1: Plain Text Notation

This is the most straightforward way to describe a domain using everyday language.

Example: For the function \(f(x) = \frac{1}{x-3}\)
  • Domain: All real numbers except x = 3
  • Explanation: We cannot divide by zero, so x cannot equal 3 (since that would make the denominator zero).
Example: For the function \(g(x) = \sqrt{x-5}\)
  • Domain: All real numbers greater than or equal to 5
  • Explanation: We cannot take the square root of a negative number, so \(x-5\) must be non-negative, meaning \(x \geq 5\).

Level 2: Set Notation

This notation uses set-builder notation with curly brackets to describe the domain mathematically.

Key symbols:

  • \(\{x \mid ...\}\) means "the set of all x such that..."
  • \(\mathbb{R}\) represents all real numbers
  • \(\neq\) means "not equal to"
  • \(\geq\) means "greater than or equal to"
  • \(\leq\) means "less than or equal to"
Example: For the function \(f(x) = \frac{1}{x-3}\)
  • Domain: \(\{x \mid x \in \mathbb{R}, x \neq 3\}\)
  • Read as: "The set of all x such that x is a real number and x is not equal to 3"
Example: For the function \(g(x) = \sqrt{x-5}\)
  • Domain: \(\{x \mid x \geq 5\}\)
  • Read as: "The set of all x such that x is greater than or equal to 5"

Level 3: Interval Notation

This notation uses intervals to describe ranges of numbers on the number line.

Key symbols:

  • \((a, b)\) and \(]a, b[\) means all numbers between a and b, not including a and b (open interval)
  • \([a, b]\) means all numbers between a and b, including a and b (closed interval)
  • \([a, b)\) and \([a, b[\) means all numbers between a and b, including a but not b
  • \(\infty\) represents infinity (always use not including with infinity)
  • \(\cup\) means "union" - combines two or more intervals
Example: For the function \(f(x) = \frac{1}{x-3}\)
  • Domain: \((-\infty, 3) \cup (3, \infty)\)
  • Read as: "All numbers from negative infinity to 3 (not including 3), union with all numbers from 3 (not including 3) to positive infinity"
  • Meaning: All real numbers except 3
Example: For the function \(g(x) = \sqrt{x-5}\)
  • Domain: \([5, \infty[\)
  • Read as: "All numbers from 5 (including 5) to positive infinity"
  • Meaning: All real numbers greater than or equal to 5
Example: For the function \(h(x) = \frac{1}{(x-2)(x+4)}\)
  • Domain: \((-\infty, -4) \cup (-4, 2) \cup (2, \infty)\)
  • Read as: "All numbers less than -4, union with all numbers between -4 and 2, union with all numbers greater than 2"
  • Meaning: All real numbers except -4 and 2

Quick Reference: Common Domain Restrictions

Function Type Restriction Reason
\(\frac{1}{x-a}\) \(x \neq a\) Cannot divide by zero
\(\sqrt{x-a}\) \(x \geq a\) Cannot take square root of negative numbers
\(\frac{1}{\sqrt{x-a}}\) \(x > a\) Cannot divide by zero AND cannot have negative under square root
\(\log(x-a)\) \(x > a\) Logarithm only defined for positive numbers

Domains shown as number lines

Help Video

This video is from The Organic Chemistry Tutor.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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