![DEFINITION: Operations on FunctionsIf f (x) and g (x) are two functions, thenSum and Difference ( f + g ) ( x ) = f(x) + g(x) Product( f g ) ( x ) = [ f(x) ] [ g(x) ] Quotient( f / g ) ( x ) = f(x) / g(x) d) Composite ( f ◦ g ) ( x ) = f (g(x))](https://image.slidesharecdn.com/7-functions-110908064649-phpapp02/85/7-functions-22-320.jpg)
7 functions
- 1. FunctionsPrepared by:Teresita P. Liwanag - Zapanta
- 2. OBJECTIVESdistinguish functions and relations
- 3. identify domain and range of a function/relationevaluate functions/relations.perform operation on functions/relations
- 4. graph functions/relationsDEFINITIONRelation is referred to as any set of ordered pair.Conventionally, It is represented by the ordered pair ( x , y ). x is called the first element or x-coordinate while y is the second element or y-coordinate of the ordered pair.
- 5. Ways of Expressing a Relation4. Graph1. Set notation5. Mapping2. Tabular form3. Equation
- 6. Example: Express the relation y = 2x;x= 0,1,2,3 in 5 ways..1. Set notation (a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) } or (b) S = { (x , y) such that y = 2x, x = 0, 1, 2, 3 }2. Tabular form
- 7. y554321x5-4-2135-5-14-32-5-1-2-3-4-5-53. Equation: y = 2x5. Mapping4. Graphxy●00●12●2463
- 8. DEFINITION: Domain and RangeAll 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.Example: Identify the domain and range of the following relations. 1.) S = { ( 4, 7 ),( 5, 8 ),( 6, 9 ),( 7, 10 ),( 8, 11 ) } Answer : D: { 4,5,6,7,8} R:{7,8,9,10,11}
- 9. 2.) S = { ( x , y ) s. t. y = | x | ; x R } Answer: D: all real nos.R: all real nos. > 03) y = x 2 – 5 Answer. D: all real nos. R: all real nos. > -5 4) | y | = x Answer: D: all real nos. > 0 R: all real nos.
- 10. g) Answer: D: all real nos. except -2 R: all real nos. except 2 5. Answer : D: all real nos. > –1 R: all real nos. > 06.Answer:D: all real nos. < 3 R: all real nos. except 07.
- 11. Exercises: Identify the domain and range of the following relations. 1. {(x,y) | y = x 2 – 4 }2. 5. 7. y = 25 – x 2y = | x – 7 |6. 4. 3. 8. y = (x 2 – 3) 29. 10.
- 12. PROBLEM SET #5-1FUNCTIONSIdentify the domain and range of the following relations.
- 13. Definition: FunctionA function is a special relation such that every first element is paired to a unique second element.
- 14. It is a set of ordered pairs with no two pairs having the same first element.FunctionsOne-to-one and many-to-one functions Consider the following graphsandEach value of x maps to only one value of y . . .Each value of x maps to only one value of y . . .and each y is mapped from only one x.BUT manyother x values map to that y.
- 15. is an example of a many-to-one functionis an example of a one-to-one function FunctionsOne-to-one and many-to-one functions Consider the following graphsandOne-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.
- 16. Example: Identify which of the following relations are functions.a) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) } b) S = { ( x , y ) s. t. y = | x | ; x R }c) y = x 2 – 5 d) | y | = x e)f)
- 17. DEFINITION: Function NotationLetters like f , g , h and the likes are used to designate functions.
- 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. The notation f ( x ) is read as “ f of x”.EXAMPLE: Evaluate each function value1. If f ( x ) = x + 9 , what is the value of f ( x 2 ) ?2. If g ( x ) = 2x – 12 , what is the value of g (– 2 )?3. If h ( x ) = x 2 + 5 , find h ( x + 1 ).If f(x) = x – 2 and g(x) = 2x2 – 3 x – 5 , Find: a) f(g(x)) b) g(f(x))
- 20. Piecewise Defined FunctionA piecewise defined function is defined by different formulas on different parts of its domain. Example:if x<0 if
- 21. Piecewise Defined FunctionEXAMPLE: Evaluate the piecewise function at the indicated values.if x<0 f(-2), f(-1), f(0), f(1), f(2)ifififif f(-5), f(0), f(1), f(5)
- 22. DEFINITION: Operations on FunctionsIf f (x) and g (x) are two functions, thenSum and Difference ( f + g ) ( x ) = f(x) + g(x) Product( f g ) ( x ) = [ f(x) ] [ g(x) ] Quotient( f / g ) ( x ) = f(x) / g(x) d) Composite ( f ◦ g ) ( x ) = f (g(x))
- 23. Example :1. Given f(x) = 11– x and g(x) = x 2 +2x –10 evaluate each ofthe following functionsf(-5) g(2)(f g)(5)(f - g)(4)f(7)+g(x)g(-1) – f(-4)(f ○ g)(x)(g ○ f)(x)(g ○ f)(2)(f○ g)
- 26. DEFINITION: Graph of a Function If f(x) is a function, then its graph is the set of all points (x,y) in the two-dimensional plane for which (x,y) is an ordered pair in f(x)One way to graph a function is by point plotting.
- 27. We can also find the domain and range from the graph of a function.
- 28. Example: Graph each of the following functions.6.
- 29. Graph of piecewise defined functionThe graph of a piecewise function consists of separate functions.Example: Graph each piecewise function.ififififif
- 30. Plot the points in the coordinate planeyx-21
- 31. Plot the points in the coordinate planeyx-21
- 32. Graph of absolute value function.Recall thatififUsing the same method that we used in graphingpiecewise function, we note that the graph of f coincides with the line y=x to the right of the y axisand coincides with the line y= -x the left of the y-axis.
- 33. Example: Graph each of the follow functions.y = | x – 7 |y = x-| x - 2 |1. 4.
- 34. Plot the points in the coordinate planeyx-21
- 36. Definition: Greatest integer function.The greatest integer function is defined by greatest integer less than or equal to xExample: 130-410-1102021
- 37. Definition: Least integer function.The least integer function is defined byleast integer greater than or equal to xExample: 240-3210212131
- 38. Graph of greatest integer function.Sketch the graph of
- 39. Plot the points in the coordinate planeyx-21
- 40. Graph of least integer function.Sketch the graph of
- 41. yPlot the points in the coordinate planex1-2