Write The Domain In Interval Notation

The domain of a function represents the set of all possible input values (often x-values) for which the function is defined and produces a real number output. Expressing this domain using interval notation provides a concise and standardized way to represent the range of permissible x-values. Mastering the skill of writing domains in interval notation is fundamental in calculus, analysis, and various branches of mathematics.

Understanding Interval Notation

Interval notation uses brackets and parentheses to denote whether the endpoints of an interval are included or excluded, respectively. Let's break down the basics:

• Brackets [ ]: Indicate that the endpoint is included in the interval.
• Parentheses ( ) : Indicate that the endpoint is excluded from the interval.
• Infinity ∞ and Negative Infinity -∞: Always used with parentheses because infinity is not a specific number and cannot be included.
• Union Symbol ∪: Used to combine multiple intervals.

Here are some examples to illustrate:

• [a, b]: Represents all real numbers x such that a ≤ x ≤ b.
• (a, b): Represents all real numbers x such that a < x < b.
• [a, b): Represents all real numbers x such that a ≤ x < b.
• (a, b]: Represents all real numbers x such that a < x ≤ b.
• [a, ∞): Represents all real numbers x such that x ≥ a.
• (-∞, b]: Represents all real numbers x such that x ≤ b.
• (-∞, ∞): Represents all real numbers, i.e., the entire real number line.

Steps to Determine and Write the Domain in Interval Notation

1. Identify Potential Restrictions: Look for situations that could lead to undefined outputs. Common restrictions include:

   • Division by Zero: The denominator of a fraction cannot be zero.
   • Square Roots (or any even root) of Negative Numbers: The radicand (the expression inside the root) must be non-negative.
   • Logarithms of Non-Positive Numbers: The argument of a logarithm must be positive.
   • Other Function-Specific Restrictions: Some functions, like tan(x) or sec(x), have inherent restrictions.

2. Solve for the Restrictions:

   • For division by zero: Set the denominator equal to zero and solve for x. These values are excluded from the domain.
   • For square roots: Set the expression inside the square root greater than or equal to zero and solve for x.
   • For logarithms: Set the argument of the logarithm greater than zero and solve for x.

3. Express the Domain in Interval Notation: Based on the restrictions, write the domain using intervals. If there are multiple intervals, use the union symbol (∪) to combine them.

4. Visualize on a Number Line (Optional but Helpful): Sketching the domain on a number line can help visualize the intervals and avoid errors.

Examples with Detailed Explanations

Let's work through several examples to solidify your understanding.

Example 1: Linear Function

• f(x) = 3x + 5

  • Restrictions: There are no restrictions. Linear functions are defined for all real numbers.
  • Domain: (-∞, ∞)

Example 2: Polynomial Function

• g(x) = x³ - 2x² + x - 7

  • Restrictions: Polynomial functions are defined for all real numbers.
  • Domain: (-∞, ∞)

Example 3: Rational Function (Division by Zero)

• h(x) = 1 / (x - 2)

  • Restrictions: The denominator cannot be zero.
  • Solve: x - 2 = 0 => x = 2
  • Domain: All real numbers except x = 2. In interval notation: (-∞, 2) ∪ (2, ∞)

Example 4: Rational Function (Division by Zero - More Complex)

• k(x) = (x + 1) / (x² - 9)

  • Restrictions: The denominator cannot be zero.
  • Solve: x² - 9 = 0 => (x - 3)(x + 3) = 0 => x = 3, x = -3
  • Domain: All real numbers except x = 3 and x = -3. In interval notation: (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)

Example 5: Square Root Function

• m(x) = √(x + 4)

  • Restrictions: The expression inside the square root must be non-negative.
  • Solve: x + 4 ≥ 0 => x ≥ -4
  • Domain: All real numbers greater than or equal to -4. In interval notation: [-4, ∞)

Example 6: Square Root Function (More Complex)

• n(x) = √(5 - 2x)

  • Restrictions: The expression inside the square root must be non-negative.
  • Solve: 5 - 2x ≥ 0 => -2x ≥ -5 => x ≤ 5/2 (Remember to flip the inequality when dividing by a negative number!)
  • Domain: All real numbers less than or equal to 5/2. In interval notation: (-∞, 5/2]

Example 7: Rational Function with a Square Root

• p(x) = √(x - 1) / (x - 3)

  • Restrictions:
    • The expression inside the square root must be non-negative: x - 1 ≥ 0 => x ≥ 1
    • The denominator cannot be zero: x - 3 = 0 => x = 3
  • Combine Restrictions: x must be greater than or equal to 1, but x cannot be equal to 3.
  • Domain: [1, 3) ∪ (3, ∞)

Example 8: Logarithmic Function

• q(x) = ln(x - 5)

  • Restrictions: The argument of the natural logarithm must be positive.
  • Solve: x - 5 > 0 => x > 5
  • Domain: All real numbers greater than 5. In interval notation: (5, ∞)

Example 9: Logarithmic Function (More Complex)

• r(x) = log₂(4 - x)

  • Restrictions: The argument of the logarithm must be positive.
  • Solve: 4 - x > 0 => -x > -4 => x < 4
  • Domain: All real numbers less than 4. In interval notation: (-∞, 4)

Example 10: Function with Multiple Logarithms

• s(x) = ln(x + 2) + ln(3 - x)

  • Restrictions: Both arguments of the natural logarithms must be positive.
  • Solve:
    • x + 2 > 0 => x > -2
    • 3 - x > 0 => -x > -3 => x < 3
  • Combine Restrictions: x must be greater than -2 and less than 3.
  • Domain: (-2, 3)

Example 11: Absolute Value in the Denominator

• t(x) = 1 / |x - 4|

  • Restrictions: The denominator cannot be zero.
  • Solve: |x - 4| = 0 => x - 4 = 0 => x = 4
  • Domain: All real numbers except x = 4. In interval notation: (-∞, 4) ∪ (4, ∞)

Example 12: Combination of Square Root and Absolute Value

• u(x) = √( |x| - 1 )

  • Restrictions: The expression inside the square root must be non-negative.
  • Solve: |x| - 1 ≥ 0 => |x| ≥ 1
  • Implications of Absolute Value: This means x ≥ 1 or x ≤ -1.
  • Domain: (-∞, -1] ∪ [1, ∞)

Example 13: Piecewise Function

• f(x) =

  • x + 1, if x < 0

  • x², if 0 ≤ x ≤ 2

  • 4, if x > 2

  • Restrictions: Piecewise functions require careful consideration of the defined intervals. In this case, each piece is defined on its respective interval without any further restrictions like division by zero or square roots.

  • Domain: Since the intervals cover all real numbers, the domain is (-∞, ∞)

Example 14: Trigonometric Function - Tangent

• v(x) = tan(x) = sin(x) / cos(x)

  • Restrictions: The cosine function in the denominator cannot be zero.
  • Solve: cos(x) = 0 => x = π/2 + kπ, where k is an integer. This means x cannot be π/2, 3π/2, -π/2, etc.
  • Domain: We need to exclude all values of the form π/2 + kπ. This is more complex to express in interval notation in a single line, but we can represent it as the union of infinitely many intervals: ... ∪ (-3π/2, -π/2) ∪ (-π/2, π/2) ∪ (π/2, 3π/2) ∪ (3π/2, 5π/2) ∪ ... A more concise, but less explicit, representation would be to describe the values that are excluded.

Example 15: Trigonometric Function - Sine and Cosine

• w(x) = sin(x) or w(x) = cos(x)

  • Restrictions: Sine and cosine functions are defined for all real numbers.
  • Domain: (-∞, ∞)

Common Mistakes to Avoid

• Forgetting Restrictions: Always meticulously check for potential restrictions before writing the domain.
• Incorrect Inequality Direction: Pay close attention to the direction of inequalities when solving for restrictions, especially when multiplying or dividing by negative numbers.
• Using Brackets Instead of Parentheses (or vice versa): Ensure you use the correct notation to indicate whether endpoints are included or excluded.
• Not Considering All Intervals: Make sure to account for all possible intervals, especially when dealing with multiple restrictions.
• Incorrectly Combining Intervals with the Union Symbol: Double-check that the intervals are correctly joined using the union symbol (∪).

Advanced Scenarios and Considerations

• Functions with Multiple Restrictions: When a function has multiple restrictions (e.g., a square root in the denominator), you must consider all of them simultaneously. The domain will be the intersection of the valid intervals for each restriction.

• Implicit Domains: Sometimes the domain of a function is not explicitly stated. In such cases, you must determine the implicit domain, which is the set of all real numbers for which the function is defined based on its formula.

• Contextual Domains: In some real-world applications, the domain may be further restricted by the context of the problem. For example, if x represents the number of items produced, the domain cannot include negative numbers.

Importance of Understanding Domains

Understanding domains is crucial for several reasons:

• Function Evaluation: It ensures you only input valid values into a function, preventing errors and undefined results.
• Graphing Functions: The domain determines the portion of the x-axis over which the function's graph exists.
• Calculus: Domains are essential for finding derivatives, integrals, and limits.
• Mathematical Modeling: In applied mathematics, domains represent the realistic and meaningful values for the variables in a model.

Practice Exercises

To further solidify your understanding, try determining the domain of the following functions and expressing them in interval notation:

1. f(x) = √(x² - 4)
2. g(x) = (x + 2) / (x² - 2x - 3)
3. h(x) = ln(2x + 5)
4. k(x) = √(9 - x²) / (x + 1)
5. m(x) = 1 / √(x - 5)

By diligently practicing and applying these concepts, you'll become proficient in identifying and expressing domains using interval notation, a vital skill for success in mathematics and related fields. Remember to always check for potential restrictions, solve for those restrictions carefully, and express your answer clearly and accurately using interval notation. The more you practice, the more intuitive this process will become.