Questions and Answers
What is a function?
A relation that assigns exactly one output for each input.
What is the domain of the function f(x) = √9 - x²?
All real x where -3 ≤ x ≤ 3.
What is the range of the function f(x) = √9 - x²?
All real y such that 0 ≤ y ≤ 3.
What is the domain of the function y = x² for 0 ≤ x ≤ 2?
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Does the equation y² = x³ represent a function?
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The range of the function y = √x + 4 is _____
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The function f(x) = 3x - x² reaches a maximum at y = _____
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What is the range of the function f(x) = 3x - x²?
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What features should be included when sketching the graph of a function?
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Which of the following functions is defined as a function?
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Study Notes
Functions Overview
- A function is a relation that uniquely associates elements from a set of inputs (domain) with a set of outputs (range).
Vertical Line Test
- A method to determine if a curve is a function: if a vertical line intersects the graph more than once, it is not a function.
Domain of a Function
- The set of all possible input values (x-values) for which the function is defined.
Range of a Function
- The set of all possible output values (y-values) that a function can produce.
Example: Function y = √x + 4
- Domain is x ≥ 0 (since the square root of a negative number is undefined).
- Range is y ≥ 4 (the minimum value occurs when x = 0).
Example: Parabola
- Features a vertex at (3, -3).
- Domain is all real x-values.
- Range is y ≥ -3.
Example: Function f(x) = 3x - x^2
- Domain is all real x.
- Range is all real y such that y ≤ 2.25.
Specifying Domain
- Function expressed as y = x^2 for 0 ≤ x ≤ 2 includes a restricted domain.
Exercise Examples
- For f(x) = √(9 - x²):
- Domain: -3 ≤ x ≤ 3.
- Range: 0 ≤ y ≤ 3.
Function Identity
- The equation y² = x³ is not a function because it can yield multiple y-values for a single x-value.
Additional Function Examples
- Functions like y = |2x| or y = |y| = x involve absolute values and exhibit specific behaviors on the graph.
Importance of Graphing
- Sketching functions aids in visualizing domains, ranges, intercepts, and identifying whether the relations are indeed functions.
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Description
Explore the concepts of functions, including their definitions, and learn how to determine the domain and range of various functions. This quiz will also cover the Vertical Line Test and provide examples to illustrate key features. Enhance your understanding of mathematical functions through engaging questions.
