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Which of the following statements accurately defines a function?
Which of the following statements accurately defines a function?
What is the primary purpose of determining the domain of a function?
What is the primary purpose of determining the domain of a function?
Which notation represents the function where $x = 3$?
Which notation represents the function where $x = 3$?
In the context of functions, what do the terms 'independent variable' and 'dependent variable' refer to?
In the context of functions, what do the terms 'independent variable' and 'dependent variable' refer to?
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What does the notation $f(x) = x^2 - 5x + 2$ imply when substituting $x = -1$?
What does the notation $f(x) = x^2 - 5x + 2$ imply when substituting $x = -1$?
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How is the range of a function determined once the domain is known?
How is the range of a function determined once the domain is known?
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Which of the following correctly describes the relationship between ordered pairs in a relation?
Which of the following correctly describes the relationship between ordered pairs in a relation?
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What characteristic distinguishes a rectangular coordinate system?
What characteristic distinguishes a rectangular coordinate system?
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What can be inferred when a function is defined with two independent variables x and y?
What can be inferred when a function is defined with two independent variables x and y?
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If f(x, y) = x³ + xy² − 2x, what is the value of f(2, 3)?
If f(x, y) = x³ + xy² − 2x, what is the value of f(2, 3)?
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Which description accurately defines a function based on its graph?
Which description accurately defines a function based on its graph?
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What does symmetry with respect to the y-axis imply for a graph?
What does symmetry with respect to the y-axis imply for a graph?
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What effect does adding a positive constant b to each y-value in a function have?
What effect does adding a positive constant b to each y-value in a function have?
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How does the graph of y = (x + 1)² compare to y = x²?
How does the graph of y = (x + 1)² compare to y = x²?
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In a symmetric graph with respect to the x-axis, what is the key characteristic?
In a symmetric graph with respect to the x-axis, what is the key characteristic?
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Which statement about the function notation z = f(x, y) is true?
Which statement about the function notation z = f(x, y) is true?
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Study Notes
Relations
- A relation is a set of ordered pairs.
- The domain of a relation is the set of all first elements in the ordered pairs.
- The range of a relation is the set of all second elements in the ordered pairs.
Functions
- A function is a relation where each element of the domain is paired with exactly one element in the range.
- Functions can be represented by equations, rules, or tables.
- When the domain is not explicitly stated, it is determined by finding the largest set of real numbers for which the equation is defined.
- The range is determined by finding the value of the equation for each value in the domain.
- The independent variable is associated with the domain, and the dependent variable is associated with the range.
Function Notation
- The notation y = f(x) means "y equals f of x," indicating that y is a function of x.
- f(a) represents the value of y (the dependent variable) when x = a.
Rectangular Coordinate System
- A rectangular coordinate system uses two perpendicular lines (X'X and Y'Y) intersecting at a point O.
- The horizontal line is called the x-axis, and the vertical line is called the y-axis.
- Any point in this system can be represented by an ordered pair (x, y), where x is the horizontal coordinate (abscissa) and y is the vertical coordinate (ordinate).
Function of Two Variables
- z is a function of two variables, x and y, if there exists a relation where each pair of values of x and y corresponds to a value of z.
- x and y are independent variables, and z is the dependent variable.
- The notation z = f(x,y) means "z equals f of x and y."
- f(a,b) represents the value of z when x = a and y = b.
Graphs of Functions
- The graph of a function y = f(x) is the set of all points (x, y) that satisfy the equation.
- A graph represents a function if for any value of x, there's only one corresponding value of y.
Graphs of NOT Functions
- Graphs that don't represent a function have multiple y-values associated with a single x-value.
Symmetry
- A graph is symmetric with respect to the y-axis if the left half is a mirror image of the right half. This occurs when f(x) = f(-x).
- A graph is symmetric with respect to the x-axis if the bottom half is a mirror image of the top half.
Shifts
- The graph of y = f(x) is shifted upward by adding a positive constant to each y-value.
- The graph of y = f(x) is shifted downward by adding a negative constant to each y-value.
- The graph of y = f(x) + b differs from the graph of y = f(x) by a vertical shift of |b| units.
- The shift is up if b > 0, and the shift is down if b < 0.
- The graph of y = (x + a)² is shifted a units to the left compared to the graph of y = x².
- The graph of y = (x - a)² is shifted a units to the right compared to the graph of y = x².
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Description
This quiz covers fundamental concepts of relations and functions in mathematics. It explores definitions, domain, range, and function notation, providing a solid foundation for understanding these key topics. Test your knowledge and improve your grasp of these essential mathematical principles.