Relations_and_Functions_Detailed. pptx

Mathematics

1. Relations and Functions Mathematics in the Modern World

2. What is a Relation? A relation is a set of ordered pairs (x, y) Defines a connection between elements of two sets Can be represented as: {(x₁, y₁), (x₂, y₂), ...} Example: {(1, 2), (2, 3), (3, 4)}

3. Real-Life Examples of Relations Student and their grades Countries and their capitals Employees and their salaries

4. Ways to Represent a Relation

5. Domain and Range of 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 a Function? 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 Identify a 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 Range in Functions Domain: All possible inputs Range: All possible outputs Example: f(x) = x², Domain: All real numbers, Range: y ≥ 0

11. Different Representations of Functions 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 Involves terms with x raised to different powers Example: f(x) = x³ - 2x² + x - 5

15. Rational and Exponential Functions Rational: f(x) = (x² + 1)/(x - 1) Exponential: f(x) = a^x

16. Special Types of Functions 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 If f(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 Relation is a Function Method 1: Check for repeated inputs with different outputs Method 2: Use the Vertical Line Test

23. Graphing Functions Plot points from a table of values Connect the points smoothly for continuous functions

24. Summary of Key Points Relations vs Functions Domain and Range Types and Representations Applications

25. Practice Problems Determine whether 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