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7 functions

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7 functions

1. 1. Functions<br />Prepared by:<br />Teresita P. Liwanag - Zapanta<br />
2. 2. OBJECTIVES<br /><ul><li>distinguish functions and relations
3. 3. identify domain and range of a function/relation</li></ul>evaluate functions/relations.<br /><ul><li>perform operation on functions/relations
4. 4. graph functions/relations</li></li></ul><li>DEFINITION<br />Relation is referred to as any set of ordered pair.<br />Conventionally, It is represented by the ordered pair <br />( x , y ). x is called the first element or x-coordinate <br />while y is the second element or y-coordinate of the <br />ordered pair. <br />
5. 5. Ways of Expressing a Relation<br />4. Graph<br />1. Set notation<br />5. Mapping<br />2. Tabular form<br />3. Equation<br />
6. 6. Example: Express the relation y = 2x;x= 0,1,2,3<br /> in 5 ways.<br />.<br />1. Set notation<br /> (a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) } or<br /> (b) S = { (x , y) such that y = 2x, x = 0, 1, 2, 3 }<br />2. Tabular form <br />
7. 7. y<br />5<br />5<br />4<br />3<br />2<br />1<br />x<br />5<br />-4<br />-2<br />1<br />3<br />5<br />-5<br />-1<br />4<br />-3<br />2<br />-5<br />-1<br />-2<br />-3<br />-4<br />-5<br />-5<br />3. Equation: y = 2x<br />5. Mapping<br />4. Graph<br />x<br />y<br />●<br />0<br />0<br />●<br />1<br />2<br />●<br />2<br />4<br />6<br />3<br />
8. 8. DEFINITION: Domain and Range<br />All the possible values of x is called the domain and all the possible values of y is called the range. In a set of ordered pairs, the set of first elements and second elements of ordered pairs is the domain and range, respectively.<br />Example: Identify the domain and range of the following <br /> relations. <br />1.) S = { ( 4, 7 ),( 5, 8 ),( 6, 9 ),( 7, 10 ),( 8, 11 ) } <br />Answer : D: { 4,5,6,7,8} R:{7,8,9,10,11}<br />
9. 9. 2.) S = { ( x , y ) s. t. y = | x | ; x  R } <br />Answer: D: all real nos.R: all real nos. > 0<br />3) y = x 2 – 5 <br />Answer. D: all real nos. R: all real nos. > -5<br /> 4) | y | = x <br />Answer: D: all real nos. > 0 R: all real nos. <br />
10. 10. g) <br />Answer: <br /> D: all real nos. except -2<br /> R: all real nos. except 2 <br />5.<br /> Answer : <br /> D: all real nos. > –1 <br /> R: all real nos. > 0<br />6.<br />Answer:<br />D: all real nos. < 3 <br />R: all real nos. except 0<br />7.<br />
11. 11. Exercises: Identify the domain and range of the <br /> following relations. <br />1. {(x,y) | y = x 2 – 4 }<br />2. <br />5. <br />7. y = 25 – x 2<br />y = | x – 7 |<br />6. <br />4. <br />3. <br />8. y = (x 2 – 3) 2<br />9. <br />10. <br />
12. 12. PROBLEM SET #5-1<br />FUNCTIONS<br />Identify the domain and range of the following relations. <br />
13. 13. Definition: Function<br /><ul><li>A function is a special relation such that every first element is paired to a unique second element.
14. 14. It is a set of ordered pairs with no two pairs having the same first element.</li></li></ul><li>Functions<br />One-to-one and many-to-one functions <br />Consider the following graphs<br />and<br />Each value of x maps to only one value of y . . .<br />Each value of x maps to only one value of y . . .<br />and each y is mapped from only one x.<br />BUT manyother x values map to that y.<br />
15. 15. is an example of a many-to-one function<br />is an example of a one-to-one function <br />Functions<br />One-to-one and many-to-one functions <br />Consider the following graphs<br />and<br />One-to-many is NOT a function. It is just a relation. Thus a function is a relation but a relation could never be a function. <br />
16. 16. Example: Identify which of the following relations are functions.<br />a) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) } <br />b) S = { ( x , y ) s. t. y = | x | ; x  R }<br />c) y = x 2 – 5 <br />d) | y | = x <br />e)<br />f)<br />
17. 17. DEFINITION: Function Notation<br /><ul><li>Letters like f , g , h and the likes are used to designate functions.
18. 18. When we use f as a function, then for each x in the domain of f , f ( x ) denotes the image of x under f .
19. 19. The notation f ( x ) is read as “ f of x”.</li></li></ul><li>EXAMPLE: Evaluate each function value<br />1. If f ( x ) = x + 9 , what is the value of f ( x 2 ) ?<br />2. If g ( x ) = 2x – 12 , what is the value of g (– 2 )?<br />3. If h ( x ) = x 2 + 5 , find h ( x + 1 ).<br />If f(x) = x – 2 and g(x) = 2x2 – 3 x – 5 ,<br /> Find: a) f(g(x)) b) g(f(x)) <br />
20. 20. Piecewise Defined Function<br />A piecewise defined function is defined by different formulas on different parts of its domain. <br />Example:<br />if x<0 <br />if<br />
21. 21. Piecewise Defined Function<br />EXAMPLE: Evaluate the piecewise function at the <br /> indicated values.<br />if x<0 <br /> f(-2), f(-1), f(0), f(1), f(2)<br />if<br />if<br />if<br />if<br /> f(-5), f(0), f(1), f(5)<br />
22. 22. DEFINITION: Operations on Functions<br />If f (x) and g (x) are two functions, then<br />Sum and Difference<br /> ( f + g ) ( x ) = f(x) + g(x) <br />Product<br />( f g ) ( x ) = [ f(x) ] [ g(x) ] <br />Quotient<br />( f / g ) ( x ) = f(x) / g(x) <br />d) Composite<br /> ( f ◦ g ) ( x ) = f (g(x)) <br />
23. 23. Example :1. Given f(x) = 11– x and g(x) = x 2 +2x –10 <br /> evaluate each ofthe following functions<br />f(-5) <br />g(2)<br />(f g)(5)<br />(f - g)(4)<br />f(7)+g(x)<br />g(-1) – f(-4)<br />(f ○ g)(x)<br />(g ○ f)(x)<br />(g ○ f)(2)<br />(f○ g)<br />
24. 24.  <br />
25. 25.  <br />
26. 26. DEFINITION: Graph of a Function <br /><ul><li>If f(x) is a function, then its graph is the set of all points </li></ul> (x,y) in the two-dimensional plane for which (x,y) is an <br /> ordered pair in f(x)<br /><ul><li>One way to graph a function is by point plotting.
27. 27. We can also find the domain and range from the</li></ul> graph of a function. <br />
28. 28. Example: Graph each of the following functions.<br />6.<br />
29. 29. Graph of piecewise defined function<br />The graph of a piecewise function consists of separate functions.<br />Example: Graph each piecewise function.<br />if<br />if<br />if<br />if<br />if<br />
30. 30. Plot the points in the coordinate plane<br />y<br />x<br />-2<br />1<br />
31. 31. Plot the points in the coordinate plane<br />y<br />x<br />-2<br />1<br />
32. 32. Graph of absolute value function.<br />Recall that<br />if<br />if<br />Using the same method that we used in graphing<br />piecewise function, we note that the graph of f <br />coincides with the line y=x to the right of the y axis<br />and coincides with the line y= -x the left of the y-axis. <br />
33. 33. Example: Graph each of the follow functions.<br />y = | x – 7 |<br />y = x-| x - 2 |<br />1. <br />4. <br />
34. 34. Plot the points in the coordinate plane<br />y<br />x<br />-2<br />1<br />
35. 35.  <br />
36. 36. Definition: Greatest integer function.<br />The greatest integer function is defined by <br />greatest integer less than or equal to x<br />Example: <br />1<br />3<br />0<br />-4<br />1<br />0<br />-1<br />1<br />0<br />2<br />0<br />2<br />1<br />
37. 37. Definition: Least integer function.<br />The least integer function is defined by<br />least integer greater than or equal to x<br />Example: <br />2<br />4<br />0<br />-3<br />2<br />1<br />0<br />2<br />1<br />2<br />1<br />3<br />1<br />
38. 38. Graph of greatest integer function.<br />Sketch the graph of<br />
39. 39. Plot the points in the coordinate plane<br />y<br />x<br />-2<br />1<br />
40. 40. Graph of least integer function.<br />Sketch the graph of<br />
41. 41. y<br />Plot the points in the coordinate plane<br />x<br />1<br />-2<br />