Domain And Range Of A Graph Interval Notation

Understanding the domain and range of a graph is fundamental to grasping the behavior and characteristics of functions. This article will explore how to determine the domain and range of a graph, with a specific focus on expressing these sets using interval notation. Whether you're a student just starting with functions or someone looking to refresh their knowledge, this guide will provide you with the tools and understanding you need.

What are Domain and Range?

The domain of a function is the set of all possible input values (usually x-values) for which the function is defined. Think of it as all the x-values that "work" in the function without causing any undefined results (like division by zero or taking the square root of a negative number).

The range, on the other hand, is the set of all possible output values (usually y-values) that the function produces when you plug in the valid x-values from the domain. It's the set of all y-values that the function "hits."

Why Use Interval Notation?

Interval notation is a concise and standardized way to represent sets of numbers, especially intervals of real numbers. It's particularly useful for expressing domains and ranges, which are often continuous sets of values. Using interval notation helps avoid ambiguity and provides a clear, unambiguous representation of the set.

Understanding Interval Notation

Before we dive into finding the domain and range from a graph, let's review the basics of interval notation:

• (a, b): This represents an open interval, meaning all numbers between a and b, excluding a and b. In other words, a < x < b. The parentheses indicate that the endpoints are not included.
• [a, b]: This represents a closed interval, meaning all numbers between a and b, including a and b. In other words, a ≤ x ≤ b. The square brackets indicate that the endpoints are included.
• (a, b]: This is a half-open (or half-closed) interval, including b but excluding a. a < x ≤ b.
• [a, b): This is also a half-open interval, including a but excluding b. a ≤ x < b.
• (-∞, b): All numbers less than b (excluding b). x < b. We always use a parenthesis with infinity because infinity isn't a specific number that can be included.
• (-∞, b]: All numbers less than or equal to b. x ≤ b.
• (a, ∞): All numbers greater than a (excluding a). x > a.
• [a, ∞): All numbers greater than or equal to a. x ≥ a.
• (-∞, ∞): Represents all real numbers.

Key Symbols:

• ( and ): Parentheses indicate that the endpoint is not included.
• [ and ]: Square brackets indicate that the endpoint is included.
• ∞ (infinity): Represents a quantity without bound. Always used with a parenthesis.
• -∞ (negative infinity): Represents a quantity without a lower bound. Always used with a parenthesis.
• ∪: Union symbol. Used to combine two or more intervals. For example, (0, 2] ∪ (5, ∞) means all numbers between 0 and 2 (including 2), and all numbers greater than 5.

Finding the Domain from a Graph

To determine the domain from a graph, follow these steps:

1. Visualize the Projection: Imagine shining a light down onto the x-axis. The shadow cast by the graph represents the domain.

2. Identify the Leftmost and Rightmost Points: Find the smallest and largest x-values that the graph covers.

3. Consider Endpoints:

   • Closed Circle (●): If the graph includes a closed circle at an endpoint, the x-value at that point is included in the domain. Use a square bracket "[" or "]" in the interval notation.
   • Open Circle (○): If the graph has an open circle at an endpoint, the x-value is not included in the domain. Use a parenthesis "(" or ")" in the interval notation.
   • Arrow (→): An arrow indicates that the graph continues indefinitely in that direction. If the arrow points to the left, the domain extends to negative infinity (-∞). If the arrow points to the right, the domain extends to positive infinity (∞).

4. Write in Interval Notation: Express the domain using interval notation based on the leftmost and rightmost points and whether the endpoints are included or excluded.

5. Discontinuities: Pay close attention to any breaks, holes (open circles), or vertical asymptotes in the graph. These points are not included in the domain and may require using the union symbol (∪) to combine multiple intervals.

Examples:

• Example 1: Line Segment

  Imagine a straight line segment that starts at the point (1, 2) with a closed circle and ends at the point (5, 4) with a closed circle.

  • Leftmost x-value: 1 (included)
  • Rightmost x-value: 5 (included)
  • Domain: [1, 5]

• Example 2: Parabola

  Consider a parabola that opens upwards. Let's say it extends infinitely to the left and right.

  • Leftmost x-value: -∞
  • Rightmost x-value: ∞
  • Domain: (-∞, ∞)

• Example 3: Rational Function (with Vertical Asymptote)

  Imagine a rational function with a vertical asymptote at x = 3. The graph exists to the left and right of the asymptote, approaching it but never touching it.

  • The graph exists for all x-values except x = 3.
  • Domain: (-∞, 3) ∪ (3, ∞)

• Example 4: A Graph with a Hole

  Suppose you have a graph that's a continuous line from x = -2 to x = 4, but there's a hole (open circle) at x = 1.

  • The graph exists for all x between -2 and 4, except for x = 1.
  • Domain: [-2, 1) ∪ (1, 4]

Finding the Range from a Graph

The process for finding the range is similar to finding the domain, but we focus on the y-values instead:

1. Visualize the Projection: Imagine shining a light horizontally onto the y-axis. The shadow cast by the graph represents the range.

2. Identify the Lowest and Highest Points: Find the smallest and largest y-values that the graph covers.

3. Consider Endpoints:

   • Closed Circle (●): If the graph includes a closed circle at an endpoint, the y-value at that point is included in the range. Use a square bracket "[" or "]" in the interval notation.
   • Open Circle (○): If the graph has an open circle at an endpoint, the y-value is not included in the range. Use a parenthesis "(" or ")" in the interval notation.
   • Arrow (↑): An arrow pointing upwards indicates that the graph continues indefinitely upwards, extending to positive infinity (∞).
   • Arrow (↓): An arrow pointing downwards indicates that the graph continues indefinitely downwards, extending to negative infinity (-∞).

4. Write in Interval Notation: Express the range using interval notation based on the lowest and highest points and whether the endpoints are included or excluded.

5. Discontinuities & Horizontal Asymptotes: Pay attention to any breaks, holes (open circles), or horizontal asymptotes. These can affect the range.

Examples:

• Example 1: Line Segment

  Using the same line segment as before (starts at (1, 2) and ends at (5, 4), both with closed circles):

  • Lowest y-value: 2 (included)
  • Highest y-value: 4 (included)
  • Range: [2, 4]

• Example 2: Parabola Opening Upwards

  Consider a parabola opening upwards with its vertex (minimum point) at (2, -1).

  • Lowest y-value: -1 (included)
  • Highest y-value: ∞
  • Range: [-1, ∞)

• Example 3: Horizontal Line

  Imagine a horizontal line at y = 3.

  • The only y-value the graph ever takes is 3.
  • Range: [3, 3] (which can also be written simply as {3})

• Example 4: A Graph with a Hole

  Suppose you have a graph that ranges from y = -3 to y = 5, but there's a hole at y = 1. The interval includes -3 and 5.

  • The range includes all y-values from -3 to 5, except for 1.
  • Range: [-3, 1) ∪ (1, 5]

• Example 5: Rational Function with Horizontal Asymptote

  Imagine a rational function with a horizontal asymptote at y = 2. The graph approaches this line as x goes to infinity and negative infinity, but never quite touches it. The graph does cover all y-values below the asymptote, going down to negative infinity.

  • The graph exists for all y-values less than 2.
  • Range: (-∞, 2)

Common Mistakes to Avoid

• Confusing Domain and Range: Always remember that domain refers to x-values and range refers to y-values.
• Forgetting Open/Closed Circles: Pay close attention to whether endpoints are included (closed circle/bracket) or excluded (open circle/parenthesis).
• Ignoring Discontinuities: Don't forget to exclude points where the function is undefined (vertical asymptotes, holes). Use the union symbol (∪) to combine intervals if necessary.
• Incorrectly Using Infinity: Always use a parenthesis with infinity (∞) and negative infinity (-∞).
• Reading the Graph Incorrectly: Carefully identify the minimum and maximum x and y values that the graph actually covers.
• Not understanding the function: It is important to try and understand the behavior of the function, as this will help you identify the domain and range. For example, you should know that the square root function, $\sqrt{x}$ , is not defined for x < 0.

Advanced Considerations

• Piecewise Functions: For piecewise functions, you need to consider each piece separately when determining the domain and range. Combine the results using the union symbol (∪).
• Functions with Restricted Domains: Some functions have inherent domain restrictions (e.g., square root functions, logarithmic functions). Be mindful of these restrictions when analyzing the graph. For example, $\sqrt{x-2}$ has a domain of [2, ∞) because the expression inside the square root must be non-negative.
• Transformations: Understanding how transformations (shifts, stretches, reflections) affect a function's graph can help you quickly determine the domain and range. For example, shifting a parabola to the right doesn't change its range, but it does change its domain if the original parabola had a restricted domain.

Examples combining domain and range in interval notation

Let's look at some more complex examples that require careful consideration of the graph:

Example 1: A piecewise function

Imagine a graph that consists of two parts:

• For x < 0, the graph is a line segment going from (-4, -2) (closed circle) to (0, 1) (open circle).
• For x ≥ 0, the graph is a parabola opening upwards with its vertex at (2, 0).

Domain:

• The first part covers (-4, 0).
• The second part covers [0, ∞).
• Since the second part includes 0, the domain is the union of these two intervals: [-4, 0) ∪ [0, ∞) = [-4, ∞)

Range:

• The first part covers [-2, 1).
• The second part covers [0, ∞).
• The range is the union of these two intervals: [-2, 1) ∪ [0, ∞) = [-2, ∞)

Example 2: A rational function with a hole and a horizontal asymptote

Suppose a rational function has a vertical asymptote at x = 1, a hole at x = 3, and a horizontal asymptote at y = 2. The graph approaches y = 2 as x goes to positive and negative infinity.

Domain:

• All real numbers except x = 1 and x = 3.
• Domain: (-∞, 1) ∪ (1, 3) ∪ (3, ∞)

Range:

• The graph gets arbitrarily close to y = 2, but never touches it, meaning 2 is not in the range. It extends downwards to -∞. It also continues above the y = 2 line. Since we don't have the equation of the line we can only say the range will be from (-∞, 2) ∪ (2, ∞). If we had the equation we could determine how low and high the y values will go on either side of the vertical asymptote.

Example 3: A square root function with a shift and reflection

Consider the function y = -√(x + 3).

Domain:

• The expression inside the square root must be non-negative: x + 3 ≥ 0 => x ≥ -3.
• Domain: [-3, ∞)

Range:

• The basic square root function y = √x has a range of [0, ∞).
• The negative sign reflects the graph across the x-axis, so the range becomes (-∞, 0].

Conclusion

Determining the domain and range of a graph and expressing them using interval notation is a vital skill in mathematics. By understanding the definitions of domain and range, mastering interval notation, and carefully analyzing the graph, you can accurately identify these sets. Remember to pay attention to endpoints, discontinuities, and any inherent restrictions of the function. With practice, you'll become proficient at extracting this important information from any graph you encounter.