FUNCTIONS. Prepared by: Ms Sonia Tan

Transcription

1 FUNCTIONS

2 FUNCTION A function is a pairing that assigns to each element of the set X eactly one element of the set Y. Using Diagrams X Y 1 a b 3 c 4 d X Y a b c X Y 1 a b 3 c One to One Many to One One to Many Function Not a Function (only a relation)

3 Using Ordered pairs {(1, a), (, b), (3, d), (4, c)} {(1, a), (1, b), (, b), (3, c)} {(1, a), (, a), (3, b), (4, b)} A relation is any set of ordered pairs. A function is a set of ordered pairs such that no two distinct ordered pairs have the same first coordinate. REAL LIFE EXAMPLES State whether a function or not. 1. A student in a math class paired with his ID number. {(Annette, 13), (Judy, 049), (Sonia, 54)}. A person paired with the type of car he or she has. { (Howard, Toyota), (Howard, BMW), (Janet, Honda)}

4 Using Tables Time Population Using Equations An equation of the form y = f() where each value of, we get eactly one value of y. Eamples: y = + 4 y 5

5 Using Graphs Graph of a function f is the set of all points (,y) on the plane that satisfy the equation y= f(). y = y y y y y 0 1 A Function Not a Function

6 Vertical Line Test for Functions If any vertical line intersects the graph of an equation at eactly one point, then the equation defines y as a function of. y y

7 Eercises: State whether a function or not using graphs 1. y. y 1 3. y 1 4. y 3

8 1. y. y y y function not a function relation

9 3. y 1 4. y 3 y y y function function

10 FUNCTION VALUES Given functions f(), f (a) means the value of the function f at = a. It is read as f of a E. f () 3 Evaluate following: f (h) f () f ( )

11 OPERATIONS ON FUNCTIONS Given functions f() and g(), we can have the Sum: f+g () = f() + g() Difference: f-g () = f() - g() Product: f g () f() g() Quotient: f g () f() g() Composition of functions: f g() f(g()) g f() EXAMPLE: Given f() = 5 g() = + 6 g(f())

12 DOMAIN AND RANGE Domain = the set of all first elements of the ordered pairs Range = the set of all second elements of the ordered pairs Eample: {( -1, 1), ( 0, 0), (, 4), ( 4, 16)} D = { -1, 0,, 4 } R = { 1, 0, 4, 16 } Give the domain and range of {(1, 5), (3, 8), (5, 11), (7, 8), (9, -4)}

13 For mathematical equation y = f() Domain = the set of all feasible values of for which y is defined Range = the set of all possible resulting values of y or images of under the f y = f() = 3 +4 D f : R R f : R y

14 y = h() = D f : all R such that 0 or R f : y 0

15 y = h() = 4 D f : all R such that or R f : y 0

16 y = h() = D f : all R such that R f : y

17 LINEAR FUNCTIONS f() = a + b where a and b are real numbers Eamples: f() = + 5 a =, b = 5 f() = -4 a = -4, b = 0 f() = 10 a = 0, b = 10 The graph of a linear function is a straight line. D f : R R f : R

18 f() = a + b + c and. a 0 QUADRATIC FUNCTIONS where a, b and c are real numbers The graph of quadratic function : If a > 0, parabola upward. If a < 0, parabola downward. V(h,k) V(h,k)

19 f() = a + b + c f() = a( - h) + k where h b a and k = f(h) V = (h, k) is called the verte of the parabola. = h V(h,k) Highest point D f : R V(h,k) R f Lowest point : y k D f : R = h R f : y k

20 Eamples: f() = a = 1, b =, c = -8 h b 1 k = f(-1) = (-1) + (-1) -8 = -9 a V = (-1, -9) f() = ( + 1) - 9 V = (-1, -9) D f : R : y 9 R f

21 Eamples: f() = a = -1, b = 1, c = 1 9 V(, 4 ) f () ( 1 ) 9 4 V( 1 9, 4 ) D f : R R f : y 9 4

22 A Polynomial function in variable of degree n is of the form n n1 P() a a a a a f() f() k n n1 where ai's R, a n is anon negativeint eger. n 0 1 and Constant function - polynomial of degree 0. a b Linear function - polynomial of degree 1. f() a b c Quadratic function - polynomial of degree f() g() polynomial of degree 4 h() 3 g and h are NOT POLYNOMIAL FUNCTIONS

23 PIECE-WISE DEFINED FUNCTIONS f() 1 For y = - + if if 1 1 When = 1, y = 0 (1, 0) When = 0, y = (0, ) For y = + 1 When = 1, y = (1, ) y = - + y = + 1 When =, y = 5 (, 5) D f : R : y 0 R f

24 g() 3 if if if 1 1 y = for < -1 y = - 3 for -1 < For y = y = y = When =, y = (, ) When = 4, y = 4 (4, 4) y = -3 D g R g : R : y 3 or y 3 [, )

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