3) (g o f) (x) = (f o g) (x) g (f (x) ) = f (g (x) ) g (2x + p) = f (3x + 120) 3(2x + p) + 120 = 2(3x + 120) + p 6x + 3p + 120 = 6x + 240 + p 3p - p = 6x - 6x + 240 - 120 2p = 120 p = 120 / 2 p = 60
4) (f o g o h) (x) = f ( g o h) (x) = f ( g (2x)) = f ((2x)² - 1) = f (4x² - 1) = 2 + 4x² + 1 = 4x² + 3 (f o g o h) (2) = 4 .2² + 3 = 4 . 4 + 3 = 16 + 3 = 19
f(x-1) = (x-1)/(2x- 1)
f(x) = (x+1 -1)/ {2(x+1) - 1)
f(x) = (x) /(2x + 1)
g(x-1) = 3x -19
g(x) = 3(x+1) - 19
g(x) = 3x - 16
(f+g)(x) = x/(2x+1) + 3x -16
(f+g)(x) = { x + 3x(2x+1) - 16(2x+1)}/(2x+1)
f+g(x) = ( x+ 6x² + 3x - 32x -16 )/(2x+1)
(f+g)(x) = (6x² - 28x - 16)/(2x+1)
3)
f(x) = 2x+p, g(x) = 3x +120
gof(x)= fog(x)
3(2x+p) + 120 = 2(3x+120) + p
6x + 3p + 120 = 6x + 240 + p
2p= 120
p = 60
4) h(2) = 2 (2)= 4
g(4)= 4² -1 = 15
f(15)= 2 + 15 = 17
fogoh(2) = fog(4)= f(15) = 17
5)
f⁻¹ (x) = (x-1)/5 --> f(x) = 5x + 1
g⁻¹(x) = (3-x)/2 --> g(X) = 3- 2x
(gof)⁻¹(6) = x --> fog(x) = 6
5(3-2x) + 1 = 6
3 -2x = 1
2x = 2
x = 1
6) (fog)⁻¹(n) =1
gof(1) = n
g { f(1)} = n
g(5)= n
n = 3 -2(5)
n = 3-10
n = -7
g (x + 1) = 3x - 19
g (x) = 3 (x + 1) - 19
= 3x + 3 - 19
= 3x - 16
(f + g) (x) = f (x) + g (x)
= [(x - 2) / (2x - 3)] + 3x - 16
= [ (x - 2) + (2x - 3) (3x - 16) ] / (2x - 3)
= (x - 2 + 6x² - 41x + 48) / (2x - 3)
= (6x² - 40x - 46) / (2x - 3)
3) (g o f) (x) = (f o g) (x)
g (f (x) ) = f (g (x) )
g (2x + p) = f (3x + 120)
3(2x + p) + 120 = 2(3x + 120) + p
6x + 3p + 120 = 6x + 240 + p
3p - p = 6x - 6x + 240 - 120
2p = 120
p = 120 / 2
p = 60
4) (f o g o h) (x) = f ( g o h) (x)
= f ( g (2x))
= f ((2x)² - 1)
= f (4x² - 1)
= 2 + 4x² + 1
= 4x² + 3
(f o g o h) (2) = 4 .2² + 3
= 4 . 4 + 3
= 16 + 3
= 19
5) (g o f)^ - 1(x) = f^-1 o g^-1 (x)
= f^- 1(g^- 1) (x)
= f^- 1 ((3 - x)/2)
= [{(3 - x)/2} - 1] / 5
= [(3 - x - 2) / 2] / 5
= (1 - x) / 10
soal no.6 sama dengan soal no.5