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.

This quiz explores the concepts of domain and range in continuous graphs. Through set builder notation, you'll learn how to represent the possible inputs and outputs for various functions. Test your understanding with examples and practical applications.