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Questions and Answers
What is the domain of the function f(x) = √(x - 5)?
What is the domain of the function f(x) = √(x - 5)?
What is the domain of the function g(x) = 1/(x² - 9)?
What is the domain of the function g(x) = 1/(x² - 9)?
Which of the following is the correct interval notation for all real numbers?
Which of the following is the correct interval notation for all real numbers?
What is the domain of the function h(x) = log₃(2x + 1)?
What is the domain of the function h(x) = log₃(2x + 1)?
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What is the domain of the function k(x) = (x + 2)/(x² + 4)?
What is the domain of the function k(x) = (x + 2)/(x² + 4)?
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What is the domain of the function f(x) = 1 / (x - 2)?
What is the domain of the function f(x) = 1 / (x - 2)?
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What is the domain of the function g(x) = √(x + 3)?
What is the domain of the function g(x) = √(x + 3)?
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What is the domain of the function h(x) = log₂(x - 1)?
What is the domain of the function h(x) = log₂(x - 1)?
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What is the domain of the function f(x) = x² - 5x + 6?
What is the domain of the function f(x) = x² - 5x + 6?
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Consider the function f(x) = x + 1, and g(x) = √x. What is the domain of the composite function f(g(x))?
Consider the function f(x) = x + 1, and g(x) = √x. What is the domain of the composite function f(g(x))?
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If a graph of a function does not include any x-values less than -5, what is the most accurate description of its domain?
If a graph of a function does not include any x-values less than -5, what is the most accurate description of its domain?
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Which of the following functions has a domain that excludes the value x = 3?
Which of the following functions has a domain that excludes the value x = 3?
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What is the domain of the function f(x) = x^4 + 2x^2 - 7?
What is the domain of the function f(x) = x^4 + 2x^2 - 7?
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Flashcards
Composite Function's Domain
Composite Function's Domain
The set of inputs allowed for both functions in a composite function.
Interval Notation
Interval Notation
A way to express the domain of functions using intervals.
Domain of f(x) = 1/(x-2)
Domain of f(x) = 1/(x-2)
The function's inputs must avoid making the denominator zero.
Domain of f(x) = √(x+3)
Domain of f(x) = √(x+3)
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Implied Domain in Real-World Contexts
Implied Domain in Real-World Contexts
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Domain of a function
Domain of a function
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Restrictions due to fractions
Restrictions due to fractions
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Square roots restrictions
Square roots restrictions
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Logarithmic functions domain
Logarithmic functions domain
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Even roots restrictions
Even roots restrictions
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Linear functions domain
Linear functions domain
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Finding domain from a graph
Finding domain from a graph
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Study Notes
Defining the Domain
- The domain of a function is the set of all possible input values (often x) for which the function is defined. It essentially describes the input values that produce a valid output.
Types of Restrictions
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Fractions: If a function includes a fraction, the denominator cannot be zero. Exclude any x values that make the denominator zero.
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Square Roots: Functions with square roots require the value inside the radical to be non-negative. Exclude any x that produces a negative value inside the square root.
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Logarithms: The argument of a logarithmic function must be positive. The part inside the log must be greater than zero.
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Even Roots: For functions with even roots (e.g., fourth root, sixth root), the expression inside the root must satisfy the non-negative condition for real outputs.
Determining the Domain in Different Function Forms
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Linear functions: Linear functions (like y = mx + b) generally have no domain restrictions, except for specified restrictions. The domain is typically all real numbers.
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Polynomial functions: Polynomial functions (like y = x² + 2x + 1) generally have no domain restrictions, unless otherwise noted. The domain is usually all real numbers.
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Rational functions: A rational function's denominator cannot equal zero. Find the excluded x-values by solving the denominator equal to zero.
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Radical functions: For square root functions or any functions with even roots, the value underneath the radical must be non-negative. This is found by solving the inequality where the expression underneath the radical is greater than or equal to zero (or zero).
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Logarithmic functions: Logarithms (like y = log₂(x)) require the argument to be strictly positive. The argument must be greater than zero; solve the inequality.
Finding the Domain from a Graph
- Visual Inspection: A graph's domain is the set of x-values the graph covers. Trace horizontally along the graph, and the x-values it spans constitute the domain.
Composite Functions
- Composite function domain: To find the domain of a composite function, consider the restrictions of both component functions. The composite function's domain is limited by inputs that work in both constituent functions.
Interval Notation
- Interval notation expresses the domain.
- (-∞, ∞) represents all real numbers.
- [a, b] represents the numbers from a to b, including a and b.
- (a, b) represents the numbers from a to b, excluding a and b.
Examples
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Example 1: f(x) = 1/(x-2): The denominator cannot be zero, so x ≠ 2. The domain is (-∞, 2) U (2, ∞).
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Example 2: f(x) = √(x+3): The expression inside the square root must be non-negative. Solving x + 3 ≥ 0 gives x ≥ -3. The domain is [-3, ∞).
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Example 3: f(x) = log₂(x-1): The argument must be positive. Solving x - 1 > 0 yields x > 1. The domain is (1, ∞).
Important Considerations
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Extraneous solutions: When solving inequalities, be wary of extraneous solutions that might not be valid in the original problem.
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Context: In real-world applications, functions often have implied domains. For example, if x represents time, then x is typically nonnegative.
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Description
This quiz focuses on understanding the domain of functions, including key restrictions such as fractions, square roots, logarithms, and even roots. Test your knowledge on what values can be used as inputs for different types of functions without resulting in mathematical errors.
