Mathematics MW Relation-and-Function.pptx

Mathematics in Modern World

1. Relation and Function

2. Objectives: Define relation and function; Identify different kinds of relation; Determine the function in a relation; and Evaluate and perform operations on functions.

3. Relation and Function Relations and Functions ” are the most important topics in algebra and these are the two different words having different meanings mathematically. In an ordered pair is represented as (INPUT, OUTPUT):  The relation shows the relationship between INPUT and OUTPUT. Whereas, a  function is a relation which derives one OUTPUT for each given INPUT. (x,y)

4. In an ordered pair we have: Domain It is a collection of the first values in the ordered pair (Set of all input (x) values). Range It is a collection of the second values in the ordered pair (Set of all output (y) values). Example: In the relation, {(-2, 3), (4 , 5), (6, -5), (-2, 5)}, The elements in the domain are {-2, 4, 6} The elements in the range are {-5, 3, 5 }. Note : Don’t consider duplicates while writing the domain and range

5. In mathematics relation is a subset of the Cartesian product. Or simply, a bunch of points (ordered pairs). In other words, the relation between the two sets is defined as the collection of ordered pairs , in which the ordered pair is formed by the object from each set. Example: {(-2, 1), (4, 3), (7, -3)}, usually written in set notation form with curly brackets .

6. Relation Representation

7. Kinds of Relation One to One One to Many Many to One 1 7 2 3 5 1 -3 2 -5 3 5 -6 -7 5 -3 1

8. What is a Function? A function is a relation which describes that there should be only one output for each input (or) we can say that a special kind of relation (a set of ordered pairs), which follows a rule i.e. every x-value should be associated with only one y-value is called a function . For example: Domain Range -1 -3 1 3 3 9 2 9 function

9. Which of the relation is a function? One to One One to Many Domain Range 2 1 3 5 7 Domain Range 3 1 3 -3 5 2 5 -5 function not a function

10. Many to One Domain Range 3 3 2 3 -1 3 -2 4 function

11. Note: All functions are relations, but not all relations are functions.

12. How to determine a graph whether a function or not? To determine a graph whether a function or not, we use the “vertical line test”. The  vertical line test  is a method that is used to determine whether a given relation is a function or not. The approach is rather simple. Draw a vertical line cutting through the graph of the relation, and then observe the points of intersection . If a vertical line intersects the graph in all places  at exactly one point , then the relation is a  function .

13. Determine the following is a function or not. 1. Graph of y = x + 1 2. Graph of y = x 2 – 2. function function

14. 3. Graph of y = x 3 4. Graph of x = y 2 function not a function

15. 5. Graph of x 2 + y 2 = 9. 6. Graph of x = y 3 – y + 2 not a function not a function

16. Naming a Function To name a function, we use letters in the alphabet in lower case. The most commonly used letters are f, g, h to name a function. 1.) y = x 2 – x - 2 2.) y = 3c + 8 3.) y = m 2 + 6m + 5 If you choose f to name #1, then you have f (x) = y. f(x) can be read as f of x. why x? it is because that’s the variable you’ve seen in the right side of the equation. 1.) f(x) = x 2 – x - 2 2.) g(c) = 3c + 8 3.) h(m) = m 2 + 6m + 5

17. Evaluation of Function To evaluate a function is just simply substitute the value of the variable and then simplify . Given the following functions: f(x) = 3x - 8 g(x) = 4x + 1 h(x) = x ² - 2x + 5 j(x) = (6x + 5)/7 find: 1.) f(5) What is f(5)? Because the name of a function is f so we take; f (x) = 3x - 8 Substitute 5 as value of x in f(x). f (5) = 3(5) – 8 Then simplify. =15 – 8 f(5) = 7

18. f(x ) = 3x - 8 g(x) = 4x + 1 h(x) = x ² - 2x + 5 j(x) = (6x + 5 )/7 2.) h(3) h(x) = x ² - 2x + 5 h(3) = (3) ² - 2(3) + 5 = 9 – 6 + 5 = 3 + 5 h(3) = 8 3.) g(0.5) g(x) = 4x + 1 g(0.5) = 4(0.5) + 1 = 2 + 1 g(0.5) = 3

19. f(x ) = 3x - 8 g(x) = 4x + 1 h(x) = x ² - 2x + 5 j(x) = (6x + 5 )/7 4.) j(5) j(x) = (6x + 5)/7 j(5) = (6(5) + 5)/7 = (30 + 5)/7 = 35/7 j(5) = 5 5.) f(2x – 1) f(x) = 3x – 8 f(2x – 1) = 3(2x -1) – 8 = 6x - 3 – 8 f(2x – 1) = 6x -11

20. Operations on Functions Here are the following operations on functions: 1.) Addition (f + g) (x) = f(x) + g(x ) example: f(x) = 3x – 4 g(x) = 5x + 1 What is (f + g) (x)? (f + g) (x) = f(x) + g(x) Substitute the given function to the formula and be sure you put a parenthesis on it. (f +g) (x) = (3x – 4) + (5x + 1) Then simplify. ( multiply + sign to (5x +1) ) = 3x – 4 + 5x + 1 = 3x + 5x – 4 + 1 (f + g) (x) = 8x - 3

21. 2.) Subtraction (f – g) (x) = f(x) – g(x) example: f(x) = 3x – 4 g(x) = 5x + 1 What is (f – g) (x)? (f – g) (x) = f(x) – g(x) Substitute the given function to the formula and be sure you put a parenthesis on it. (f – g) (x) = (3x – 4) – (5x + 1) Then simplify. ( multiply – sign to (5x + 1) ) =3x – 4 - 5x – 1 = 3x – 5x - 4 – 1 ( f – g) (x) = -2x - 5

22. 3.) Multiplication ( fg ) (x) = f(x)  g(x) example: f(x) = 3x – 4 g(x) = 5x + 1 What is ( fg ) (x)? ( fg ) (x) = f(x)  g(x) Substitute the given function to the formula and be sure you put a parenthesis on it. ( fg ) (x) = (3x – 4)  (5x + 1) Then simplify. ( you can use the foil method ) = 15x ² + 3x – 20x – 4 ( fg ) (x) = 15x² - 17 x - 4

23. 4.) Division (f/g) (x) = f(x) / g(x) example: f(x) = x ² + 2x + 1 g(x) = x + 1 9/3 3(3)/3 3/1=3 What is ( f/g ) (x)? ( fg ) (x) = f(x) / g(x) Substitute the given function to the formula and be sure you put a parenthesis on it. ( f/g ) (x) = (x ² + 2x + 1 ) / (x + 1) Then simplify. ( you can use factoring method –express the numerator by its factors ) = {(x + 1) (x + 1)} / (x + 1) Cancel one group of (x + 1) in both numerator and denominator. (f/g) (x) = x + 1

24. 5.) Composite of a Function ( f o g ) (x) = f (g(x)) example: f(x) = 3x – 4 g(x) = 5x + 1 What is (f o g) (x)? (f o g) (x) = f (g(x)) The value of x in f(x) is the g(x) f (x) = 3x – 4 f(g(x)) = 3(g(x)) - 4 Substitute the function g(x) to the value of x in f(x) and be sure you put a parenthesis on it. f(5x + 1) = 3(5x + 1) - 4 Then simplify. = 15x + 3 – 4 [f(g)] (x ) = 15x -1