Domain and Range from Graph Calculator

Domain and Range from Graph Calculator

Domain (X-values)

Range (Y-values)

Results:

Domain:

Range:

Understanding Domain and Range from Graphs

When analyzing a function or a relation represented graphically, two fundamental concepts are its domain and range. These describe the set of all possible input values (x-values) and output values (y-values) that the graph covers.

What is Domain?

The domain of a graph refers to all the possible x-values for which the function or relation is defined. To find the domain from a graph, you look at the graph's extent along the horizontal (x-axis). Imagine "squishing" the entire graph onto the x-axis; the portion of the x-axis that is covered by the graph represents its domain.

• Leftmost Point: Identify the smallest x-value the graph reaches.
• Rightmost Point: Identify the largest x-value the graph reaches.
• Interval Notation: The domain is typically expressed using interval notation, which uses parentheses () for values that are not included (open circles, asymptotes, or infinity) and square brackets [] for values that are included (closed circles or points the graph passes through).
• Infinity: If the graph extends indefinitely to the left or right (indicated by an arrow), the domain includes negative infinity (-∞) or positive infinity (∞), respectively. Infinity is always represented with a parenthesis.

What is Range?

The range of a graph refers to all the possible y-values (output values) that the function or relation can produce. To find the range from a graph, you look at the graph's extent along the vertical (y-axis). Imagine "squishing" the entire graph onto the y-axis; the portion of the y-axis that is covered by the graph represents its range.

• Lowest Point: Identify the smallest y-value the graph reaches.
• Highest Point: Identify the largest y-value the graph reaches.
• Interval Notation: Similar to the domain, the range is expressed using interval notation, with parentheses () for excluded values and square brackets [] for included values.
• Infinity: If the graph extends indefinitely downwards or upwards, the range includes negative infinity (-∞) or positive infinity (∞), respectively.

How to Use the Calculator

This calculator helps you determine the domain and range of a graph by inputting its observed boundaries. You'll specify the leftmost, rightmost, lowest, and highest points the graph reaches, and whether those points are included or if the graph extends to infinity.

1. For Domain (X-values):
   • Enter the numerical value for the graph's leftmost point on the x-axis. If the graph extends infinitely to the left, check "Extends to Negative Infinity".
   • Indicate if the minimum x-value is included (e.g., a closed circle) or excluded (e.g., an open circle or asymptote).
   • Do the same for the maximum x-value (rightmost point).
2. For Range (Y-values):
   • Enter the numerical value for the graph's lowest point on the y-axis. If the graph extends infinitely downwards, check "Extends to Negative Infinity".
   • Indicate if the minimum y-value is included or excluded.
   • Do the same for the maximum y-value (highest point).
3. Click "Calculate Domain & Range" to see the results in interval notation.

Examples of Domain and Range Determination:

Example 1: A Line Segment

Consider a line segment starting at (-2, 1) with a closed circle and ending at (3, 4) with an open circle.

• Domain Inputs:
  • Minimum X-value: -2, Included: Yes
  • Maximum X-value: 3, Included: No
• Range Inputs:
  • Minimum Y-value: 1, Included: Yes
  • Maximum Y-value: 4, Included: No
• Calculator Output:
  • Domain: [-2, 3)
  • Range: [1, 4)

Example 2: A Parabola Opening Upwards

Consider a parabola with its vertex at (1, -3) that opens upwards, extending infinitely.

• Domain Inputs:
  • Extends to Negative Infinity (Left): Yes
  • Extends to Positive Infinity (Right): Yes
• Range Inputs:
  • Minimum Y-value: -3, Included: Yes
  • Extends to Positive Infinity (Up): Yes
• Calculator Output:
  • Domain: (-∞, ∞)
  • Range: [-3, ∞)

Example 3: A Square Root Function

Consider a square root function starting at (0, 0) (closed circle) and extending infinitely to the right and upwards.

• Domain Inputs:
  • Minimum X-value: 0, Included: Yes
  • Extends to Positive Infinity (Right): Yes
• Range Inputs:
  • Minimum Y-value: 0, Included: Yes
  • Extends to Positive Infinity (Up): Yes
• Calculator Output:
  • Domain: [0, ∞)
  • Range: [0, ∞)