What Is Interval Notation For Domain

Interval notation offers a concise and standardized way to represent the domain and range of functions, as well as solution sets to inequalities, using intervals of real numbers. It's a fundamental concept in mathematics, especially in calculus, analysis, and pre-calculus, providing a clear and unambiguous method to define the set of all possible input values (domain) or output values (range) of a function. Understanding interval notation is crucial for accurately communicating mathematical ideas and for solving problems involving inequalities and functions.

Understanding the Basics of Interval Notation

Interval notation uses brackets and parentheses to indicate whether the endpoints of an interval are included or excluded. The notation always lists the lower bound of the interval first, followed by the upper bound, separated by a comma. Let's break down the key components:

Parentheses ( ): Indicate that an endpoint is not included in the interval. This is used for open intervals or when the endpoint is not part of the solution set. For example, (a, b) represents all real numbers between a and b, excluding a and b.
Brackets [ ]: Indicate that an endpoint is included in the interval. This is used for closed intervals or when the endpoint is part of the solution set. For example, [a, b] represents all real numbers between a and b, including a and b.
Infinity (∞) and Negative Infinity (-∞): These symbols represent unbounded intervals. Infinity is always enclosed in a parenthesis because infinity is not a real number and cannot be included as an endpoint. For example, (a, ∞) represents all real numbers greater than a, and (-∞, b] represents all real numbers less than or equal to b.
Union (∪): This symbol is used to combine two or more intervals. For example, (a, b) ∪ (c, d) represents the set of all numbers in either the interval (a, b) or the interval (c, d).

Representing Different Types of Intervals

Here's a breakdown of how to represent different types of intervals using interval notation:

Open Interval (a, b): Represents all real numbers strictly between a and b. This does not include a or b. Example: (2, 5) represents all numbers between 2 and 5, but not 2 and 5 themselves.
Closed Interval [a, b]: Represents all real numbers between a and b, including a and b. Example: [-1, 3] represents all numbers between -1 and 3, including -1 and 3.
Half-Open (or Half-Closed) Intervals: These intervals include one endpoint but not the other.
- (a, b] represents all real numbers greater than a and less than or equal to b. Example: (0, 4] represents all numbers greater than 0 and less than or equal to 4.
- [a, b) represents all real numbers greater than or equal to a and less than b. Example: [2, 6) represents all numbers greater than or equal to 2 and less than 6.
Unbounded Intervals: These intervals extend to infinity in one or both directions.
- (a, ∞) represents all real numbers greater than a. Example: (5, ∞) represents all numbers greater than 5.
- [a, ∞) represents all real numbers greater than or equal to a. Example: [-2, ∞) represents all numbers greater than or equal to -2.
- (-∞, b) represents all real numbers less than b. Example: (-∞, 1) represents all numbers less than 1.
- (-∞, b] represents all real numbers less than or equal to b. Example: (-∞, 7] represents all numbers less than or equal to 7.
- (-∞, ∞) represents all real numbers, i.e., the entire real number line.

Domain of a Function and Interval Notation

The domain of a function is the set of all possible input values (often represented by the variable x) for which the function is defined and produces a real number output. Identifying the domain is a crucial step in understanding a function's behavior and limitations. Interval notation provides a concise and accurate way to express the domain.

Common Restrictions on the Domain:

Several factors can restrict the domain of a function. Here are some common examples:

Division by Zero: A function is undefined when the denominator is zero. Therefore, any value of x that makes the denominator zero must be excluded from the domain.
Square Roots of Negative Numbers: In the realm of real numbers, you cannot take the square root of a negative number. Therefore, any value of x that results in a negative number under a square root must be excluded from the domain. This restriction applies to other even-indexed radicals as well (fourth root, sixth root, etc.).
Logarithms of Non-Positive Numbers: The logarithm of a non-positive number (zero or negative) is undefined. Therefore, any value of x that results in a non-positive number inside a logarithm must be excluded from the domain.

Steps to Determine the Domain and Express it in Interval Notation:

Identify Potential Restrictions: Look for any of the restrictions mentioned above (division by zero, square roots of negative numbers, logarithms of non-positive numbers).
Solve for the Restricted Values: Set the denominator equal to zero and solve for x to find values that must be excluded due to division by zero. Set the expression under a square root greater than or equal to zero and solve for x to find the values that are allowed under the square root. Set the expression inside a logarithm greater than zero and solve for x to find the allowed values inside the logarithm.
Write the Domain in Interval Notation: Use the information from step 2 to construct the intervals that represent the domain. Use parentheses to exclude restricted values and brackets to include allowed values. Use the union symbol (∪) to combine multiple intervals if necessary.

Examples of Finding Domains and Expressing them in Interval Notation

Let's illustrate how to find the domain of different functions and express it using interval notation:

Example 1: f(x) = 1/x

Restriction: Division by zero.
Solve: x = 0 must be excluded.
Domain in Interval Notation: (-∞, 0) ∪ (0, ∞)

Example 2: g(x) = √(x - 3)

Restriction: Square root of a negative number.
Solve: x - 3 ≥ 0 => x ≥ 3
Domain in Interval Notation: [3, ∞)

Example 3: h(x) = ln(x + 2)

Restriction: Logarithm of a non-positive number.
Solve: x + 2 > 0 => x > -2
Domain in Interval Notation: (-2, ∞)

Example 4: k(x) = (x + 1) / (x² - 4)

Restriction: Division by zero.
Solve: x² - 4 = 0 => (x - 2)(x + 2) = 0 => x = 2 or x = -2
Domain in Interval Notation: (-∞, -2) ∪ (-2, 2) ∪ (2, ∞)

Example 5: m(x) = √(4 - x²)

Restriction: Square root of a negative number.
Solve: 4 - x² ≥ 0 => x² ≤ 4 => -2 ≤ x ≤ 2
Domain in Interval Notation: [-2, 2]

Example 6: p(x) = 1 / √(x - 1)

Restriction: Division by zero AND square root of a negative number. Since the square root is in the denominator, we need the expression inside the square root to be strictly greater than zero.
Solve: x - 1 > 0 => x > 1
Domain in Interval Notation: (1, ∞)

Example 7: q(x) = ln( (x+3) / (x-1) )

Restriction: Logarithm of a non-positive number AND division by zero. We need the expression inside the logarithm to be strictly greater than zero, i.e., (x+3)/(x-1) > 0. We also need to make sure x-1 != 0, so x != 1.
Solve: To solve (x+3)/(x-1) > 0, we can use a sign chart. The critical points are x = -3 and x = 1.
- For x < -3, both (x+3) and (x-1) are negative, so (x+3)/(x-1) is positive.
- For -3 < x < 1, (x+3) is positive and (x-1) is negative, so (x+3)/(x-1) is negative.
- For x > 1, both (x+3) and (x-1) are positive, so (x+3)/(x-1) is positive.
Domain in Interval Notation: (-∞, -3) ∪ (1, ∞)

Range of a Function and Interval Notation

The range of a function is the set of all possible output values (often represented by the variable y or f(x)) that the function can produce. Determining the range can be more challenging than determining the domain, and often requires analyzing the function's behavior, including its graph, limits, and critical points. Interval notation is equally useful for expressing the range.

Finding the Range:

There's no single method for finding the range that works for all functions. Here are some common approaches:

Graphing: Graph the function and visually identify the lowest and highest y-values the function attains.
Analyzing the Function: Consider the function's properties, such as whether it has a maximum or minimum value, whether it is always positive or always negative, and whether it has any horizontal asymptotes.
Considering the Inverse Function: Find the inverse function (if it exists) and determine its domain. The domain of the inverse function is the range of the original function. This method is most effective when the inverse function is easier to analyze than the original function.

Examples of Finding Ranges and Expressing them in Interval Notation

Example 1: f(x) = x²

Analysis: The square of any real number is non-negative. The minimum value is 0 (when x=0), and the function increases without bound as x moves away from 0 in either direction.
Range in Interval Notation: [0, ∞)

Example 2: g(x) = sin(x)

Analysis: The sine function oscillates between -1 and 1.
Range in Interval Notation: [-1, 1]

Example 3: h(x) = e^x

Analysis: The exponential function is always positive. As x approaches negative infinity, e^x approaches 0, but never reaches it. As x approaches positive infinity, e^x increases without bound.
Range in Interval Notation: (0, ∞)

Example 4: k(x) = 1/(x² + 1)

Analysis: The denominator x² + 1 is always greater than or equal to 1. The maximum value of k(x) occurs when x = 0, and k(0) = 1. As x approaches positive or negative infinity, k(x) approaches 0. Since x² + 1 is always positive, k(x) is always positive.
Range in Interval Notation: (0, 1]

Example 5: m(x) = √x

Analysis: The square root function always returns a non-negative value. As x increases, √x also increases. When x = 0, √x = 0.
Range in Interval Notation: [0, ∞)

Combining Intervals with the Union Symbol (∪)

Sometimes, the domain or range of a function may consist of multiple disjoint intervals. In such cases, we use the union symbol (∪) to combine these intervals. We saw examples of this earlier when discussing domains involving rational functions.

Example: f(x) = √(x - 2) + √(5 - x)

Restriction: We need both x - 2 ≥ 0 AND 5 - x ≥ 0
Solve:
- x - 2 ≥ 0 => x ≥ 2
- 5 - x ≥ 0 => x ≤ 5
Combining the Intervals: We need x to be both greater than or equal to 2 AND less than or equal to 5. This can be written as 2 ≤ x ≤ 5.
Domain in Interval Notation: [2, 5] (This is a single interval, so we don't need the union symbol.)

Example (Revisited): k(x) = (x + 1) / (x² - 4)

Domain in Interval Notation: (-∞, -2) ∪ (-2, 2) ∪ (2, ∞) (Here, we need the union symbol to combine the three separate intervals.)

Practice Problems

To solidify your understanding of interval notation, try these practice problems:

Find the domain of f(x) = √(x + 5) / (x - 2) and express it in interval notation.
Find the domain of g(x) = ln(3 - x) and express it in interval notation.
Find the range of h(x) = x² + 3 and express it in interval notation.
Find the range of k(x) = -√x and express it in interval notation.
Find the domain of m(x) = 1 / (x² + 5x + 6) and express it in interval notation.
Find the domain of p(x) = √(9 - x²) and express it in interval notation.
Find the range of q(x) = |x| (absolute value of x) and express it in interval notation.
Find the domain of r(x) = ln(x² - 1) and express it in interval notation.

Conclusion

Interval notation is an essential tool for expressing the domain and range of functions, as well as solution sets to inequalities, in a precise and standardized manner. By understanding the meaning of parentheses, brackets, infinity, and the union symbol, you can effectively communicate mathematical concepts and solve problems involving functions and inequalities with clarity and accuracy. Mastering this notation is a crucial step towards success in calculus and higher-level mathematics. Remember to always consider potential restrictions on the domain, analyze the function's behavior, and practice applying the concepts to various examples. With consistent practice, you'll become proficient in using interval notation to confidently express mathematical ideas.