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Questions and Answers
The domain of a rational function includes all real numbers, except those that make the denominator zero.
The domain of a rational function includes all real numbers, except those that make the denominator zero.
True (A)
The range of the function f(x) = x² - 4x + 3 is from [-1, ∞).
The range of the function f(x) = x² - 4x + 3 is from [-1, ∞).
True (A)
For the function f(x) = √(x+3), the domain is restricted to x < -3.
For the function f(x) = √(x+3), the domain is restricted to x < -3.
False (B)
The range of the function f(x) = 1/(x-2) includes the value y = 0.
The range of the function f(x) = 1/(x-2) includes the value y = 0.
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The domain of a logarithmic function must include zero and negative values.
The domain of a logarithmic function must include zero and negative values.
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For a square root function, the values under the radical must be positive to determine the domain.
For a square root function, the values under the radical must be positive to determine the domain.
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Piecewise functions have a domain that can be determined separately for each piece.
Piecewise functions have a domain that can be determined separately for each piece.
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The range of a quadratic function is determined by the x-coordinate of the vertex.
The range of a quadratic function is determined by the x-coordinate of the vertex.
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Study Notes
Domain and Range of Functions
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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.
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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
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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.
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Rational functions (fractions): The denominator cannot be zero. Exclude values of x that result in a zero denominator.
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Radical functions (square roots): The expression inside the radical must be non-negative. Solve the inequality for the variable.
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Logarithmic functions: The argument of a logarithm must be positive. Solve the inequality for the variable.
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Piecewise functions: Determine the domain for each piece independently, then consider the union of these domains.
Determining Range
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Graphical approach: Graph the function. The range includes all y-values the graph touches or crosses.
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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
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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, ∞).
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Function: f(x) = √(x+3)
- Domain: x ≥ -3. Expressed as [-3, ∞).
- Range: y ≥ 0. Expressed as [0, ∞).
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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
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Understanding domain and range is crucial to understanding a function's behavior and input-output mapping.
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Domain and range information helps determine valid inputs and expected outputs in equations and models.
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Determining the domain first when working with a function can immediately identify potential limitations and constraints.
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Description
Test your understanding of the domain and range of various functions. This quiz covers real-number functions, rational functions, and radical functions, examining how to determine valid inputs and expected outputs. Check your grasp on identifying and excluding values that affect domain and range.
