General Mathematics Lecture Notes
Introduction
- Lecture for incoming Grade 11 students
- Focus on the topic of functions within General Mathematics
Key Concepts
Domain and Range
- Domain: Set of first coordinates (x-values) in a set of ordered pairs
- Range: Set of second coordinates (y-values) in a set of ordered pairs
- Example: Ordered pairs
- (1, -1), (2, -3), (0, 5), (-1, 3), (4, -5)
- Domain: {-1, 0, 1, 2, 4}
- Range: {-5, -4, -3, -1, 3, 5}
Definitions
- Relation: A set of ordered pairs
- Function: A relation where each element of the domain corresponds to exactly one element of the range
Representations of Functions
Ordered Pairs
- To determine if a set of ordered pairs is a function, ensure no x-values are repeated with different y-values.
- Example:
- Relation F: {(1,2), (2,2), (3,5), (4,5)} is a function
- Relation G: {(1,3), (1,4), (2,5), (2,6), (3,7)} is not a function (repeated x-values)
- Relation H: {1,3), (2,6), (3,9)} is a function
Mapping Diagrams
- Input (domain) and output (range) values are mapped
- If an input maps to more than one output, it's not a function
- Examples:
- Diagram 1: Inputs {10, 20, 30, 40} map to outputs {15, 25, 35, 45} - Function
- Diagram 2: Input 10 maps to both 20 and 40 - Not a function
- Diagram 3: Inputs B and C map to the same output - Function
Graphs (using Vertical Line Test)
- A graph represents a function if and only if each vertical line intersects the graph at most once
- Examples:
- Graph with a straight vertical line intersecting at one point - Function
- Graph intersected by a vertical line at two or more points - Not a function
Equations
- Linear equations (e.g., y = 2x + 1) typically represent functions
- Quadratic equations (e.g., x^2 + y - 4 = 0) can represent functions
- Rational equations (e.g., y = 2x + 1 / x - 1) need verification through graphing
- Non-function example: x^2 + y^2 = 1 when solved for y gives two values for every x
Practice Questions
Ordered Pairs
- Determine if sets of ordered pairs are functions:
- Set 1: {(-7, 4), (-8, 3), (-7, 7), (-20, 8)} - Not a function
- Set 2: {(10, 9), (-2, -16), (-6, 7), (5, 8)} - Function
- Set 3: {(-13, 4), (7, -15), (-13, 9), (6, -12)} - Not a function
- Set 4: {(6, 0), (-12, -16), (-6, 10), (20, -7)} - Function
Mapping Diagrams
- Determine if mapping diagrams represent functions:
- Diagram 1: Inputs {3}, outputs {0, 2, 4, 6} - Not a function
- Diagram 2: Inputs {X, Y, U, V, L}, outputs {3, 5, 7, 8, 10, 15} - Function
Graphs
- Using the vertical line test:
- Graph 1: Not a function
- Graph 2: Function
Equations
- Determine if equations describe a function:
- y = 2x + 1 - Function
- x^2 + y^2 = 9 - Not a function (two values for y for each x)
Conclusion
- Reviewed and practiced identifying functions from ordered pairs, mapping diagrams, graphs, and equations
- Next lesson: Determining domain and range from equations
- References and further reading provided
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