Introduction to Domain and Range

Key Concepts

• Domain: Set of all possible x-values of a function.
• Range: Set of all possible y-values of a function.

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Example 1: Determining Domain and Range from a Graph

Domain

• Lowest x value: -4
• Highest x value: 3
• Interval notation: [-4, 3] (includes -4 and 3)

Range

• Lowest y value: -5
• Highest y value: 4
• Interval notation: [-5, 4] (includes -5 and 4)

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Example 2: Another Function

Domain

• Lowest x value: -6 (open circle, not included)
• Highest x value: 6 (closed circle, included)
• Interval notation: (-6, 6]

Range

• Lowest y value: -4 (open circle, not included)
• Highest y value: 5 (closed circle, included)
• Interval notation: (-4, 5]

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Example 3: Function with Infinity

Domain

• Lowest x value: 1 (closed circle, included)
• Goes to infinity
• Interval notation: [1, ∞)

Range

• Lowest y value: 2 (closed circle, included)
• Goes to infinity
• Interval notation: [2, ∞)

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Example 4: Downward Parabola

Domain

• Goes to left (negative infinity) and right (positive infinity)
• Interval notation: (-∞, ∞)

Range

• Lowest y value: -∞
• Highest y value: 3 (closed circle, included)
• Interval notation: (-∞, 3]

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Example 5: Jump in Graph

Domain

• Lowest x value: -6 (open circle, not included)
• Highest x value: 5 (closed circle, not included)
• Interval notation: (-6, 5)
• Second part: from -2 (closed circle, included) to 5 (not included) and from 7 to infinity
• Final domain: (-6, 5) ∪ [-2, 5) ∪ [7, ∞)

Range

• Lowest y value: -4 (open circle, not included)
• Highest y value: 4 (closed circle, included)
• Notice the gap between -2 and 1
• Interval notation: (-6, -2] ∪ [1, 4]

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Example 6: Complex Function

Domain

• Lowest x value: -8 (included)
• Highest x value: -4 (not included)
• Second part: from -2 (included) to 5 (not included)
• Third part: from 7 to infinity
• Final domain: [-8, -4) ∪ [-2, 5) ∪ [7, ∞)

Range

• Lowest y value: -6 (not highest)
• Highest y value: 1
• From 2 to 5 (gap)
• Final range: [-6, 2] ∪ [5, ∞)

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Summary

• Finding domain and range involves identifying the extreme x and y values and their inclusion or exclusion.
• Use interval notation to express the domain and range clearly.