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Relations_and_Functions_Detailed. pptx
- 1. Relations and Functions Mathematicsin the Modern World
- 2. What is aRelation? • A relation is a set of ordered pairs (x, y) • Defines a connection between elements of two sets • Can be represented as: {(x , y ), (x , y ), ...} ₁ ₁ ₂ ₂ • Example: {(1, 2), (2, 3), (3, 4)}
- 3. Real-Life Examples ofRelations • Student and their grades • Countries and their capitals • Employees and their salaries
- 4. Ways to Representa Relation
- 5. Domain and Rangeof a Relation • Domain: All first elements (x-values) • Range: All second elements (y-values) • Example: Relation {(1, 2), (2, 3), (3, 4)} • Domain = {1, 2, 3}, Range = {2, 3, 4}
- 6. What is aFunction? • A function is a special type of relation • Every element in domain maps to exactly one element in range • No element in domain has more than one image in range
- 7. Examples of Functions •f(x) = 2x + 3 → Linear Function • f(x) = x² → Quadratic Function • f(x) = √x → Radical Function
- 8. Examples of Non-Functions •{(1, 2), (1, 3), (2, 4)} – Not a function because 1 maps to both 2 and 3
- 9. How to Identifya Function? • Check if any input has more than one output • Graphically: Use Vertical Line Test • If any vertical line intersects the graph more than once, it's not a function
- 10. Domain and Rangein Functions • Domain: All possible inputs • Range: All possible outputs • Example: f(x) = x², Domain: All real numbers, Range: y ≥ 0
- 11. Different Representations ofFunctions • 1. Equation Form (y = f(x)) • 2. Table of Values • 3. Mapping Diagram • 4. Graph
- 12. Linear Functions • Form:y = mx + b • Graph: Straight line • Example: y = 2x + 1
- 13. Quadratic Functions • Form:y = ax² + bx + c • Graph: Parabola • Example: y = x² + 2x + 1
- 14. Polynomial Functions • Involvesterms with x raised to different powers • Example: f(x) = x³ - 2x² + x - 5
- 15. Rational and ExponentialFunctions • Rational: f(x) = (x² + 1)/(x - 1) • Exponential: f(x) = a^x
- 16. Special Types ofFunctions • One-to-One Function • Many-to-One Function • Onto Function • Into Function
- 17. Composite Functions • Definition:(f g)(x) = f(g(x)) ∘ • Example: f(x) = x², g(x) = x + 1 • f(g(x)) = (x + 1)²
- 18. Inverse Functions • Iff(x) = y, then f⁻¹(y) = x • Example: f(x) = 2x + 3 → f⁻¹(x) = (x - 3)/2
- 19. Applications of Functions •Business: Cost and revenue functions • Science: Speed and distance relations • Technology: Algorithm performance
- 20. Real-Life Example 1 •Temperature conversion: F = (9/5)C + 32
- 21. Real-Life Example 2 •Area of a circle as a function of radius: A(r) = πr²
- 22. Checking if Relationis a Function • Method 1: Check for repeated inputs with different outputs • Method 2: Use the Vertical Line Test
- 23. Graphing Functions • Plotpoints from a table of values • Connect the points smoothly for continuous functions
- 24. Summary of KeyPoints • Relations vs Functions • Domain and Range • Types and Representations • Applications
- 25. Practice Problems • Determinewhether the following are functions: • 1. {(1, 2), (2, 3), (3, 4)} • 2. {(1, 2), (1, 3), (2, 4)} • Find domain and range for f(x) = x² - 1
- 26. End of Presentation •Thank you! • Questions and Discussion