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(1) The document defines four functions: f(x)=2x-6, g(x)=-3x+5, h(x)=x^2-1, k(x)=(2x+5)^2-1. It then defines operations on functions such as addition, subtraction, multiplication, and composition. (2) Examples are given of calculating the sum, difference, and product of two functions, as well as the composite function g∘f. The domain and range of the composite functions are discussed. (3) The inverse of a function is defined. Examples inverse functions are calculated from relations provided in the text.
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- 1. 9 (Increasing Function) T.KAINOY (Decreasing Function) (Increasing Function) x2 > x1 f (x2) > f (x1) x f (x) f(x) = 2x – 6 (Decreasing Function) x2 > x1 f (x2) < f (x1) x f (x) f(x) = -3x+5 1. f + g = {(x , y) y = f(x) + g(x)} Df+g = Df Dg
- 2. 10 T.KAINOY 2. f - g = {(x , y) y = f(x) - g(x)} Df-g = Df Dg 3. f . g = {(x , y) y = f(x).g(x)} Dfg = Df Dg 4. f = {(x , y) y = f (x ) } D f = Df g g (x ) g Dg-{x g(x) = 0} 9 f + g , f – g , fg (1) (2) (3) (4) ( ) 10 f ( x) 2 x 1 g ( x) 3 x 4 f (f g )(x) , (f g )(x) , ( fg)(x) (x) g (1) (f g )(x) = f ( x) g ( x) = (2 x 1) (3x 4) = ……………………………… (2) (f g )(x) = f ( x) g ( x) = (2 x 1) (3x 4)
- 3. 11 T.KAINOY = ……………………………… (3) ( fg)(x) = f ( x ) g ( x) = (2 x 1)(3x 4) = ………………………………………………………………………… = ………………………………………………………………………… f f ( x) (4) ( )( x ) = g g ( x) = ………………………………………………………………………… (Composite function) f g f g
- 4. 12 T.KAINOY f g gof g f fog g f fog gof f g fog g gof(x) = g(f(x)) f fog(x) = f(g(x)) 11 gof
- 5. 13 T.KAINOY gof(x) = g(f(x)) gof(1) = g(f(1)) = g(2) = 4 gof(2) = g(f(2)) = g(3) = 9 gof(3) = g(f(3)) = g(4) = 16 gof = Dgof = Rgof= 12 gof fog gof = fog = 13 gof fog Rf = R Dg = R Rf Dg ≠ gof gof(x) = g(f(x)) = g(2x+5) = (2x+5)2 – 1 = 4x2+20x +24 Rg ≥ -1 Df = R Rg Df ≠ fog fog(x) = f(g(x))
- 6. 14 T.KAINOY = f(x2-1) = 2(x2-1) + 5 = 2x2 + 3 14 gof(3) fog(1) Rf …….. Dg ……. Rf Dg ≠ gof gof(x) = g(f(x)) = = = = gof(3) = Rg …… Df ……. Rg Df ≠ fog fog(x) = f(g(x)) = = = fog(1) = 1) gof fog gof f g fog g f 2) Dgof = Df Rgof Rg
- 7. 15 T.KAINOY 3) (fog)oh = fo(goh) 4) gof f g (Inverse Function) f-1 f-1 = (y,x) (x,y) f r f f f −1 f −1 f –1 f − 1(x) = (0,3) ,(1,4) ,(2,5) = (3,0) ,(4,1) ,(5,2) = (0,3) ,(1,3) ,(2,4) = (3,0) ,(3,1) ,(4,2)
- 8. 16 T.KAINOY - - 15 f(x) = 2x – 7 (x1,y) f y = 2x1 – 7 (x2,y) f y = 2x2 – 7 2x1 – 7 = 2x2 – 7 x1 = x2 f(x) = 2x – 7 f(x) = 2x – 7 16 f(x) = x-6 + 3 f(x) = 10 f(x) = x-6 + 3 10 = x-6 + 3 7 = x-6 x – 6= 7 x = 13 -1 f(x) = x-6 + 3 f(x) = x-6 + 3 1) ( f 1 ( x)) 1 f ( x) 2) ( fof 1 )( x) ( f 1of )( x) x
- 9. 17 T.KAINOY 3) ( gof ) 1 ( x) f 1 ( x)og 1 ( x) 4) ( fof 1 )( x) ( f 1of )( x) 2 1 (x,y) f ( x, y) A B| y 2x 7 1 f ( y, x) B A | y 2x 7 2 x y f ( x, y) A B| y 2x 7 1 f ( x, y ) B A | x 2y 7 y = 1 x 7 f ( x, y ) B A| y 2 17 f ( x, y ) A B| y 3x 1 2x 5 ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………. ……………………………………………………………………………… ……………………………………………………………………………… …………………………………………………………………………….
- 10. 18 T.KAINOY ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………. ……………………………………………………………………………… ……………………………………… 18 f (x) (1,5), (2,4), (3,6), (7,9) g (x) (4,1), (2,2), (1,4), ( 2,7) 1 (f g )( x) f (x) (1,5), (2,4), (3,6), (7,9) g (x) (4,1), (2,2), (1,4), ( 2,7) g 1 ( x) (1,4), (2,2), (4,1), (7, 2) Df 1,2,3,7 Dg 1 1,2,4,7 Df Dg 1 1,2,7 (f g 1 )( x) = f ( x) g 1 ( x) = (1,......... .), (2,......... ..), (7,......... ....) = ……………………………………….. 19 f ( x) 2x 3 g ( x) x2 5 1 1 f g (5) f (x) y 2x 3 1 f ( x) x 2y 3 x 3 y 2 2 g (x) y x 5 g 1 ( x) x y2 5 y x 5
- 11. 19 T.KAINOY f 1 g 1 ( x) = f 1 ( x) g 1 ( x) = x 3 x 5 2 f 1 g 1 (5) = ……………………………………………………………… = ……………………………………………………… ……… 20 f ( x) 3 x 5 g ( x) x2 1 f 1og 1 ( x) f (x) y 3x 5 1 f ( x) x 3y 5 x 5 y 3 2 g (x) y x 1 g 1 ( x) x y2 1 y x 1 f 1og 1 ( x) = f 1 ( g 1 ( x)) = f 1 ( x 1) = x 1 5 3
- 12. 20 T.KAINOY 21 f (x) (0,5), (1,9), (2,10 ), (3,15) g (x) (3,5), ( 1,9), (2,10 ), (4,7) g 1of ( x) g 1 ( x) (5,3), (9, 1), (10 ,2), (7,4) g 1of ( x) =……………………………………………………….. 1. f g f 1 ( x) x 4 ( fog )( x) 3 x 2 2 3 f ( x) g ( x) x 4 x 4 f 1 ( x) y 3 3 f y x x y y 4 x 3 y 3x 4 f ( x) 3 x 4 ( fog )( x) 3 x 2 2 f ( g ( x)) 3x 2 2 3g ( x) 4 3 x 2 2 g (x) ………………………………..
- 13. 21 T.KAINOY g (x) ……………………………….. f ( x) g ( x) = …………………………………………………………………… ………….. = …………………………………………………………………… ………….. 2. f ( x 1) 3x 2 f ( x) g (3x 1) 2 x 8 f (0) 1 g 1 ( f ( 2)) f ( x 1) 3x 2 f ( x) f( ) = f- x 0 f (0 1) 3(0) 2 f (0) 1( ) = f (1) 0 2 1 f (1) 3 x 1 f (1 1) 3(1) 2 f (1) f (2) 3 2 3 f (2) 8 g 1 ( f (2)) g 1 (8) g (3x 1) 2 x 8 g 1 (2 x 8) 3x 1 2x 8 8 x 0 g 1 (2 x 8) g 1 (8) 3(0) 1 1 g 1 ( f (2)) 1 3. f ( x) x2 5 g ( x) x 5 (1) (gof)(5) (2) (fog)(7) (1) (gof)(5) = ……………………………………………………………………………… ……..
- 14. 22 T.KAINOY = ……………………………………………………………………………… …….. = ……………………………………………………………………………… …….. = ……………………………………………………………………………… …….. (2) (fog)(7) = ……………………………………………………………………………… …….. = ……………………………………………………………………………… …….. = ……………………………………………………………………………… …….. = ……………………………………………………………………………… …….. 4. f ( x) 2 x 1 ( fog )( x) x3 3 g (x) ……………………………………………………………………… …………………………………………
- 15. 23 T.KAINOY ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………. 5. f ( x) 3 x 2 2x 1 x 1 ,3 ,5 ……………………………………………………………………… ………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… …………………………………….