DEFINITION Relation 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.
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Ways of Expressing a Relation 4. Graph 1. Set notation 5. Mapping 2. Tabular form 3. Equation
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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
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y 5 5 4 3 2 1 x 5 -4 -2 1 3 5 -5 -1 4 -3 2 -5 -1 -2 -3 -4 -5 -5 3. Equation: y = 2x 5. Mapping 4. Graph x y ● 0 0 ● 1 2 ● 2 4 6 3
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DEFINITION: Domain and Range 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. 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}
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2.) S = { ( x , y ) s. t. y = | x | ; x R } Answer: D: all real nos.R: all real nos. > 0 3) 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.
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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. > 0 6. Answer: D: all real nos. < 3 R: all real nos. except 0 7.
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Exercises: Identify the domain and range of the following relations. 1. {(x,y) | y = x 2 – 4 } 2. 5. 7. y = 25 – x 2 y = | x – 7 | 6. 4. 3. 8. y = (x 2 – 3) 2 9. 10.
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PROBLEM SET #5-1 FUNCTIONS Identify the domain and range of the following relations.
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Definition: Function
A function is a special relation such that every first element is paired to a unique second element.
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It is a set of ordered pairs with no two pairs having the same first element.
Functions One-to-one and many-to-one functions Consider the following graphs and Each 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.
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is an example of a many-to-one function is an example of a one-to-one function Functions One-to-one and many-to-one functions Consider the following graphs and 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.
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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)
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DEFINITION: Function Notation
Letters like f , g , h and the likes are used to designate functions.
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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 .
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The notation f ( x ) is read as “ f of x”.
EXAMPLE: Evaluate each function value 1. 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))
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Piecewise Defined Function A piecewise defined function is defined by different formulas on different parts of its domain. Example: if x<0 if
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Piecewise Defined Function EXAMPLE: Evaluate the piecewise function at the indicated values. if x<0 f(-2), f(-1), f(0), f(1), f(2) if if if if f(-5), f(0), f(1), f(5)
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DEFINITION: Operations on Functions If f (x) and g (x) are two functions, then Sum 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))
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Example :1. Given f(x) = 11– x and g(x) = x 2 +2x –10 evaluate each ofthe following functions f(-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)
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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.
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We can also find the domain and range from the
graph of a function.
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Example: Graph each of the following functions. 6.
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Graph of piecewise defined function The graph of a piecewise function consists of separate functions. Example: Graph each piecewise function. if if if if if
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Plot the points in the coordinate plane y x -2 1
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Plot the points in the coordinate plane y x -2 1
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Graph of absolute value function. Recall that if if Using the same method that we used in graphing piecewise function, we note that the graph of f coincides with the line y=x to the right of the y axis and coincides with the line y= -x the left of the y-axis.
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Example: Graph each of the follow functions. y = | x – 7 | y = x-| x - 2 | 1. 4.
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Plot the points in the coordinate plane y x -2 1
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Definition: Greatest integer function. The greatest integer function is defined by greatest integer less than or equal to x Example: 1 3 0 -4 1 0 -1 1 0 2 0 2 1
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Definition: Least integer function. The least integer function is defined by least integer greater than or equal to x Example: 2 4 0 -3 2 1 0 2 1 2 1 3 1
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Graph of greatest integer function. Sketch the graph of
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Plot the points in the coordinate plane y x -2 1
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Graph of least integer function. Sketch the graph of
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y Plot the points in the coordinate plane x 1 -2