Domain and Range of Continuous Graphs

Domain and Range of Continuous Graphs

• The domain of a continuous graph represents all the possible input values (usually x-values) for which the graph is defined. Think of it as the "acceptable" inputs.
• The range of a continuous graph represents all the possible output values (usually y-values) that the graph can produce. Think of it as the possible results.

Set Builder Notation for Continuous Graphs

• Set builder notation is a way to describe a set of numbers using a rule.
  • It's a concise way to represent the domain and range of a function.

Example 1: A Straight Line

• Consider the straight line graph of the equation y = 2x + 1.
• Domain: In this case, there are no restrictions on the x-values. You can plug in any real number for x and get a corresponding y-value.
  • Domain in set builder notation: {x | x ∈ ℝ} (This means all real numbers)
• Range: Similar to the domain, there are no restrictions on the y-values. The graph will extend infinitely in both directions along the y-axis, which corresponds to any real number.
  • Range in set builder notation: {y | y ∈ ℝ}

Example 2: A Parabola

• Now consider the parabola y = x².
• Domain: Again, there are no restrictions on the x-values. You can plug in any real number for x and calculate a corresponding y-value.
  • Domain in set builder notation: {x | x ∈ ℝ}
• Range: Notice that the parabola opens upwards, meaning the smallest y-value is 0. All possible y-values are greater than or equal to 0.
  • Range in set builder notation: {y | y ≥ 0, y ∈ ℝ}

Example 3: A Square Root Function

• Consider the square root function y = √x.
• Domain: For a square root function, the input (x) inside the square root must be greater than or equal to zero to avoid imaginary numbers.
  • Domain in set builder notation: {x | x ≥ 0, x ∈ ℝ}
• Range: Since x and y are related by the square root, the smallest possible value for the square root is 0, which produces a value of 0 for y. The square root will yield positive values for all positive inputs.
  • Range in set builder notation: {y | y ≥ 0, y ∈ ℝ}

Example 4: A rational function (e.g., y = 1/x).

• Domain The denominator cannot be zero.
  • Domain in set builder notation: {x|x ≠ 0, x ∈ ℝ}
• Range The function can take many values, but it cannot be zero.
  • Range in set builder notation: {y|y ≠ 0, y ∈ ℝ}

General Guidelines for Determining Domain From a Graph

• Look for any breaks or gaps on the graph. The domain does not include any values of x for which there's no corresponding point on the graph.
• The domain is the set of all input values (x-coordinates) that the graph uses.
• Typically, a continuous graph will have a domain of all real numbers.

Key Takeaways

• Set builder notation provides a way to describe the domain and range of continuous graphs concisely.
• The domain and range are sets of numbers.
• Knowing how to find the domain and range helps to better understand how a function operates.
• The specific notation depends on the type of function (e.g., linear, quadratic, radical, rational).
• Real numbers (ℝ) are used extensively to specify the possible values.