Domain and Range of Functions Quiz

Domain and Range of Functions

• The domain of a function is the set of all possible input values (x-values) for which the function is defined. It represents the valid inputs.

• The range of a function is the set of all possible output values (y-values) that the function can produce. It represents the outputs.

Determining Domain

• Real-number functions: Functions involving only real numbers typically have a domain of all real numbers, unless restricted by division by zero or the square root of a negative number.

• Rational functions (fractions): The denominator cannot be zero. Exclude values of x that result in a zero denominator.

• Radical functions (square roots): The expression inside the radical must be non-negative. Solve the inequality for the variable.

• Logarithmic functions: The argument of a logarithm must be positive. Solve the inequality for the variable.

• Piecewise functions: Determine the domain for each piece independently, then consider the union of these domains.

Determining Range

• Graphical approach: Graph the function. The range includes all y-values the graph touches or crosses.

• Algebraic approach (depending on the function):

  • Quadratic functions: Determine the vertex. The range is all values greater than or equal to the y-coordinate of the vertex (upward-opening parabola) or less than or equal to the y-coordinate (downward-opening).
  • Maximum/Minimum Values: Analyze the function's algebraic definition to find maximum or minimum values, which help determine the range. Express the range in interval form.
  • Known Function Transformations: Transformations of known functions can assist in determining the range.

Examples

• Function: f(x) = 1/(x-2)

  • Domain: All real numbers except x = 2. Expressed as (-∞, 2) U (2, ∞).
  • Range: All real numbers except y = 0. Expressed as (-∞, 0) U (0, ∞).

• Function: f(x) = √(x+3)

  • Domain: x ≥ -3. Expressed as [-3, ∞).
  • Range: y ≥ 0. Expressed as [0, ∞).

• Function: f(x) = x² - 4x + 3

  • Domain: All real numbers.
  • Range: y ≥ -1. (The minimum value occurs at x = 2).

Interval Notation

• Parentheses ( ): Used for open intervals (values not included).
• Brackets [ ]: Used for closed intervals (values included).
• Infinity (∞) and negative infinity (-∞): Always use parentheses with infinity symbols.

Key Concepts

• Understanding domain and range is crucial to understanding a function's behavior and input-output mapping.

• Domain and range information helps determine valid inputs and expected outputs in equations and models.

• Determining the domain first when working with a function can immediately identify potential limitations and constraints.