Question

.......
1.Diketahui :
.. f(x) = 2x + 3
.. g(x) = x + 2 / x - 5
tentukan
a. nilai dari (g bundaran f) (x)
b. invers dari (g bundaran f) (x)

2. Diketahui
.. f(x) = 8x + 2 / 3 - 5x
.. g(x) = 11x + 20
tentukan :
a. nilai dari (g bundaran f)
b. invers dari (g bundaran f)


Jawab dengan benar dan tepat dan jangan asal Jawab thanks

Answer 1

No. [1] :

f(x) = 2x + 3
g(x) = ( x + 2 ) / ( x - 5 )

(a) ( g o f )(x) = g [ f(x) ]
= g ( 2x + 3 )
= ( 2x + 3 + 2 ) / ( 2x + 3 - 5 )
= ( 2x + 5 ) / ( 2x - 2 )

(b) Rumus :
y = ( ax + b ) / ( cx + d ) ---> y⁻¹ = ( -dx + b ) / ( cx - a )

( g o f )(x) = ( 2x + 5 ) / ( 2x - 2 )
( g o f )⁻¹ (x) = ( 2x + 5 ) / ( 2x - 2 ) ---> invers ( g o f )(x) tidak berubah



No. [2] :

f(x) = ( 8x + 2 ) / ( 3 - 5x )
g(x) = 11x + 20

(a) ( g o f )(x) = g [ f(x) ]
= g [ ( 8x + 2 ) / ( 3 - 5x ) ]
= 11 [ ( 8x + 2 ) / ( 3 - 5x ) ] + 20
= [ ( 88x + 22 ) / ( 3 - 5x ) ] + 20 ---> samakan penyebut
= [ ( 88x + 22 ) / ( 3 - 5x ) ] + [ 20 ( 3 - 5x ) / ( 3 - 5x ) ]
= [ ( 88x + 22 ) / ( 3 - 5x ) ] + [ ( 60 - 100x ) / ( 3 - 5x ) ]
= [ ( 88x + 22 ) + ( 60 - 100x ) ] / ( 3 - 5x ) 
= ( -12x + 82 ) / ( -5x + 3 ) 

(b) Rumus :
y = ( ax + b ) / ( cx + d ) ---> y⁻¹ = ( -dx + b ) / ( cx - a )
( g o f )(x) = ( -12x + 82 ) / ( -5x + 3 )
( g o f )⁻¹ (x) = ( -3x + 5 ) / ( -5x + 12 ) 

Answer 2

1. a) gof = g(f(x)) = 2x + 3 + 2/ 2x + 3 -5 = (2x + 5)/ (2x - 2)

b) invers dari 2x + 5/ 2x -2
y = 2x + 5/ 2x -2
2xy - 2y = 2x + 5
2xy - 2x = 5 + 2y
x(2y - 2) = 5 + 2y
x = 5 + 2y / 2y - 2
Maka invers nya adalah (5 + 2x) / (2x - 2)

2. a) g o f = g (f(x)) = 11 (8x + 2 / 3 - 5x) + 20
g o f = 88x + 22 / (3 - 5x)    + 20
samain penyebut
g o f = 88x + 22 + 60 - 100x / (3 - 5x)\
g o f = (-12x + 82) / (3 - 5x)

b) invers dari g o f
y = -12x + 82 / 3 - 5x
3y - 5xy = -12x + 82
12x - 5xy = -3y + 82
x(12 - 5y) = -3y + 82
x = -3y + 82/ (12 - 5y)
Maka invers g o f = (-3x + 82)/ (12 - 5x)

Semoga Membantu ^^