Questions and Answers
Which of the following represents the correct definition of the domain of a function?
The range of a function is the set of all possible inputs.
False
What is the formula for the area of a circle in terms of radius?
A(r) = πr^2
Match the function types with their definitions:
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For the function A(r) = πr^2, the domain is __________.
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Which of the following is a characteristic of a quadratic function?
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The ordered pair notation for a function is represented as (x, f(x)).
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What is the range of the function A(r) = πr^2?
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The image of an element x in the domain is denoted as __________.
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Which of the following statements is true regarding the linear function?
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Study Notes
Functions and Relations
- Relation: Defined as a set of ordered pairs. The first component of each pair is called the domain, while the second component is the range.
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Example Relations:
- {(1, 4), (2, 5), (3, 6), (4, 8)} is an example of a valid relation.
- {(1, a), (1, b), (1, c), (1, d)} is not a function since one domain element corresponds to multiple range elements.
Functions
- Function: A special type of relation where each domain element maps to exactly one range element.
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Examples of Functions:
- {(2, 1), (3, 1), (4, 1), (5, 1)} illustrates a function with unique outputs for distinct inputs.
Key Comparisons
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Relation vs. Function:
- A relation shows the link between sets, while a function guarantees a precise outcome for each input.
- An example of a function: Heights and names—height cannot lead to multiple names, ensuring a one-to-one mapping.
Function Types
- Identity Function: Defined as f(x) = x, where m = 1 and b = 0.
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Absolute Value Function: Defined as
- f(x) = x for x ≥ 0
- f(x) = -x for x < 0
- Piecewise Function: Contains multiple sub-functions, each applicable to specific intervals of the domain.
Exercises
- Evaluating whether relations constitute a function includes providing their domain and range.
- Examples of relation evaluation:
- {(2, 3), (4, 5), (6, 6)} is a function, while {(5, 1), (5, 2), (5, 3)} is not since one input relates to multiple outputs.
Domain and Range
- Domain: Set of all x-coordinates (inputs) in a function's graph, defined as {x | x is an element of X}.
- Range: Set of all y-coordinates (outputs) corresponding to the domain values.
- Example for a golf ball trajectory: Domain may vary; for instance, {3 ≤ x ≤ 12}, with a corresponding range {6 ≤ y ≤ 12}.
More Functions
- Linear Function: Takes the form f(x) = mx + b, where both m and b are real numbers, and at least one is non-zero.
- Constant Function: A case of a linear function where m = 0, hence f(x) = b.
- Quadratic Function: Any function expressed as f(x) = ax² + bx + c, where a ≠ 0.
Example Functions
- Area of a circle: A(r) = πr².
- Valid domain: r must be greater than 0, ensuring radius cannot be negative. Thus, the domain is (0, ∞) and the range is also (0, ∞).
Conclusion
- Understanding the distinction between relations and functions is crucial in mathematics.
- Functions can be classified into various types, and recognizing their domain and range is essential for practical applications, particularly in real-world scenarios.
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Description
Test your understanding of functions and their representations in mathematics. This quiz covers key concepts, including the definition of functions, determining if a relation represents a function, and understanding piece-wise functions. Dive into real-life applications of these mathematical concepts as well.
