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

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

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

• (f o g) x
• (g o f) x

Jawaban

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

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

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

Jawaban

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

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

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

Jawaban

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

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

Jawaban

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

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

Jawaban

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