Soal-Soal Matematika/Fungsi komposisi

Bentuk dan sifat fungsi komposisi

bentuk (f o g) x = f(g (x))

sifat:

1. (f o g) x ≠ (g o f) x
2. ((f o g) o h) x = (f o (g o h)) x
3. (f o I) x = (I o f) x

tambahan sifat-sifat dari fungsi invers.

contoh soal

1. f(x) = 6x-7, g(x) = 8x+1 serta h(x) = 5x-3. tentukan

• (f o g) x
• (g o f) x
• ((f o g) o h) x
• (f o (g o h)) x

Jawaban

$${\begin{aligned}*(f\circ g)x&=f(g(x))\\&=6(8x+1)-7\\&=48x+6-7\\&=48x-1\\*(g\circ f)x&=g(f(x))\\&=8(6x-7)+1\\&=48x-56+1\\&=48x-55\\*(f\circ g)x&=f(g(x))\\&=6(8x+1)-7\\&=48x+6-7\\&=48x-1\\((f\circ g)\circ h)x&=(f\circ g)(h(x))\\&=48(5x-3)-1\\&=240x-144-1\\&=240x-145\\*(g\circ h)x&=g(h(x))\\&=8(5x-3)+1\\&=40x-24+1\\&=40x-23\\(f\circ (g\circ h))x&=f((g\circ h)(x))\\&=6(40x-23)-7\\&=240x-138-7\\&=240x-145\\\end{aligned}}$$

2. f(x) = 6x-7, g(x) = 4x²-8x+1. tentukan

• (f o g) x
• (g o f) x

Jawaban

$${\begin{aligned}*(f\circ g)x&=f(g(x))\\&=6(4x^{2}-8x+1)-7\\&=24x^{2}-48x+6-7\\&=24x^{2}-48x-1\\*(g\circ f)x&=g(f(x))\\&=4(6x-7)^{2}-8(6x-7)+1\\&=4(36x^{2}-84x+49)-48x+56+1\\&=144x^{2}-336x+196-48x+57\\&=144x^{2}-384x+253\\\end{aligned}}$$

3. tentukan g(x) bila f(x) = 6x-7 serta

• f(g(x)) = 24x²-48x-1!
• g(f(x)) = 144x²-384x+253!

Jawaban

$${\begin{aligned}*f(g(x))&=24x^{2}-48x-1\\6(g(x))-7&=24x^{2}-48x-1\\{\frac {6(g(x))-7+7}{6}}&={\frac {24x^{2}-48x-1+7}{6}}\\g(x)&=4x^{2}-8x+1\\*g(f(x))&=144x^{2}-384x+253\\g(6x-7)&=144x^{2}-384x+253\\g({\frac {6x-7+7}{6}})&=144({\frac {x+7}{6}})^{2}-384({\frac {x+7}{6}})+253\\g(x)&={\frac {144(x^{2}+14x+49)}{36}}-{\frac {(384x+2.688)}{6}}+253\\g(x)&={\frac {24(x^{2}+14x+49)}{6}}-{\frac {(384x+2.688)}{6}}+{\frac {1.518}{6}}\\g(x)&={\frac {24x^{2}+336x+1.176-384x-2.688+1.518}{6}}\\g(x)&={\frac {24x^{2}-48x+6}{6}}\\g(x)&=4x^{2}-8x+1\\\end{aligned}}$$

4. tentukan f(x) bila g(x) = 4x²-8x+1 serta

• f(g(x)) = 24x²-48x-1!
• g(f(x)) = 144x²-384x+253!

Jawaban

$${\begin{aligned}*f(g(x))&=24x^{2}-48x-1\\f(4x^{2}-8x+1)&=24x^{2}-48x-1\\f(4x^{2}-8x+1)&=24x^{2}-48x-1+7-7\\f(4x^{2}-8x+1)&=24x^{2}-48x+6-7\\f(4x^{2}-8x+1)&=6(4x^{2}-8x+1)-7\\f(x)&=6x-7\\*g(f(x))&=144x^{2}-384x+253\\4f(x)^{2}-8f(x)+1&=144x^{2}-384x+253\\4f(x)^{2}-8f(x)+4-3&=144x^{2}-384x+256-3\\(2f(x)-2)^{2}-3&=(12x-16)^{2}-3\\(2f(x)-2)^{2}&=(12x-16)^{2}\\2f(x)-2&=12x-16\\2(f(x)-1)&=2(6x-8)\\f(x)-1&=6x-8\\f(x)&=6x-7\\\end{aligned}}$$

5 Diketahui f(x)=2x+1 dan (f ο g)(x-1) = -2x²+4x-1. Tentukan g(-2)!

Jawaban

$${\begin{aligned}(f\circ g)(x-1)&=-2x^{2}+4x-1\\f(g(x-1))&=-2x^{2}+4x-1\\{\text{misalkan}}g(x-1)&=a\\f(a)&=-2x^{2}+4x-1\\2a+1&=-2x^{2}+4x-1\\2a&=-2x^{2}+4x-2\\a&=-x^{2}+2x-1\\a&=-(x^{2}-2x+1)\\a&=-(x-1)^{2}\\g(x-1)&=a\\g(x-1)&=-(x-1)^{2}\\g(x)&=-x^{2}\\g(-2)&=-(-2)^{2}\\g(-2)&=-4\\\end{aligned}}$$

6 Diketahui f(x-4)=2x+1 dan g(x+3) = x²-5x+6. Tentukan (f ο g)^-1(9)!

Jawaban

$${\begin{aligned}(f\circ g)^{-1}(9)&=g^{-1}(f^{-1}(9))\\f(x-4)&=2x+1\\f(x)&=2(x+4)+1\\f(x)&=2x+8+1\\f(x)&=2x+9\\f^{-1}(x)&={\frac {x-9}{2}}\\g(x+3)&=x^{2}-5x+6\\g(x)&=(x-3)^{2}-5(x-3)+6\\g(x)&=x^{2}-6x+9-5x+15+6\\g(x)&=x^{2}-11x+30\\g(x)&=x^{2}-11x+{\frac {121}{4}}-{\frac {1}{4}}\\g(x)&=(x-{\frac {11}{2}})^{2}-{\frac {1}{4}}\\g^{-1}(x)&={\sqrt {x+{\frac {1}{4}}}}+{\frac {11}{2}}\\(f\circ g)^{-1}(x)&=g^{-1}(f^{-1}(x))\\g^{-1}(f^{-1}(x))&={\sqrt {{\frac {x-9}{2}}+{\frac {1}{4}}}}+{\frac {11}{2}}\\g^{-1}(f^{-1}(9))&={\sqrt {{\frac {9-9}{2}}+{\frac {1}{4}}}}+{\frac {11}{2}}\\g^{-1}(f^{-1}(9))&={\sqrt {\frac {1}{4}}}+{\frac {11}{2}}\\g^{-1}(f^{-1}(9))&={\frac {1}{2}}+{\frac {11}{2}}\\g^{-1}(f^{-1}(9))&={\frac {12}{2}}\\g^{-1}(f^{-1}(9))&=6\\\end{aligned}}$$