Podcast
Questions and Answers
What is the definition of a function?
What is the definition of a function?
Which of the following statements about the domain and range is true?
Which of the following statements about the domain and range is true?
What is the purpose of the vertical line test?
What is the purpose of the vertical line test?
In a piecewise function, what does it mean when a subfunction applies to a certain portion of the main function's domain?
In a piecewise function, what does it mean when a subfunction applies to a certain portion of the main function's domain?
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What happens when you perform operations on functions with overlapping domains?
What happens when you perform operations on functions with overlapping domains?
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Study Notes
Functions and Relations
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Definitions
- Domain: Set of first coordinates in ordered pairs.
- Range: Set of second coordinates in ordered pairs.
- Examples: For pairs (1,2), (3,4), (5,6), (7,8), Domain: {1,3,5,7}, Range: {2,4,6,8}.
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Relations
- Defined as a set of ordered pairs; includes a Domain and its Range.
- Domain refers to the set of all x-components, while the Range includes all y-components.
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Functions
- A function relates each element of the domain to exactly one element in the range.
- Example Domain: {5,10,15,20}, Range: {2,3,4,5}.
- Values arranged in ascending order.
Representation of Functions
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Functions can be represented in various forms:
- Ordered Pairs: Example: (x, y).
- Table of Values: Lists corresponding values of x and y.
- Mapping Diagram: Visual representation linking domain to range.
- Vertical Line Test: A graphical method to determine if a relation is a function; if a vertical line intersects the graph more than once, it's not a function.
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Note:
- Domain values cannot be repeated while range values may repeat.
Evaluation of Functions
- To evaluate a function, replace the variable x with a given value.
- Example: For h(x) = (6x - 3)/7, and find h(4):
- h(4) = (6*4 - 3)/7 = (24 - 3)/7 = 21/7 = 3.
- Example: For h(x) = (6x - 3)/7, and find h(4):
Piecewise Functions
- Defined by multiple sub-functions, each applies to a specific domain portion.
- Example: f(x) = {4 + x for x² - 2}.
- Calculation:
- For f(0): f(0) = 0² + 5 = 5.
- For f(5): f(5) = 5 - 2 = 3.
Operations on Functions
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Functions with overlapping domains can be:
- Added: (f + g)(x) = f(x) + g(x).
- Subtracted: (f - g)(x) = f(x) - g(x).
- Multiplied: (fg)(x) = f(x) * g(x).
- Divided: (f/g)(x) = f(x) / g(x), where g(x) ≠ 0.
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These operations apply to any functions f(x) and g(x) over their common domain.
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Description
Explore the fundamental concepts of functions and relations in this quiz. Understand the definitions of domain and range, and how to identify ordered pairs, domain sets, and range sets. Test your knowledge through examples and different representations such as tables and mappings.
