SNBT Zone : Fungsi Komposisi

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StandarSNBTHOTS

No.

Jika \((f\circ g)(x)=\dfrac{6x+3}{2x-5}\) dan g(x) = 4x − 11, maka hasil dari \(\displaystyle\intop_5^8\dfrac{f^{-1}(x-1)}{g(3)}\ dx\) adalah

1. 72 ln 2 − 3
2. 36 ln 3 − 2
3. 36 ln 2 − 6

4. 36 ln 2 − 3
5. 72 ln 3 − 2

Grup Matematika Idhamdaz

ALTERNATIF PENYELESAIAN

\(\begin{aligned} (f\circ g)(x)&=\dfrac{6x+3}{2x-5}\\[3.5pt] f(g(x))&=\dfrac{12x+6}{4x-10}\\[3.5pt] f(4x-11)&=\dfrac{3(4x-11)+39}{4x-11+1}\\[3.5pt] f(x)&=\dfrac{3x+39}{x+1}\\[3.5pt] f^{-1}(x)&=\dfrac{-x+39}{x-3}\\[3.5pt] f^{-1}(x-1)&=\dfrac{-(x-1)+39}{x-1-3}\\[3.5pt] &=\dfrac{-x+1+39}{x-4}\\[3.5pt] &=\dfrac{-x+40}{x-4} \end{aligned}\)

\(\begin{aligned} \displaystyle\intop_5^8\dfrac{f^{-1}(x-1)}{g(3)}\ dx&=\displaystyle\intop_5^8\dfrac{\dfrac{-x+40}{x-4}}{4(3)-11}\ dx\\ &=\displaystyle\intop_5^8\dfrac{\dfrac{-x+4+36}{x-4}}{12-11}\ dx\\ &=\displaystyle\intop_5^8\dfrac{-1+\dfrac{36}{x-4}}1\ dx\\ &=\displaystyle\intop_5^8\left(-1+\dfrac{36}{x-4}\right)\ dx\\ &=\left[-x+36\ln|x-4|\right]_5^8\\ &=\left[-8+36\ln|8-4|\right]-\left[-5+36\ln|5-4|\right]\\ &=\left[-8+36\ln4\right]-\left[-5+36\ln1\right]\\ &=\left[-8+36\ln2^2\right]-\left[-5-42(0)\right]\\ &=\left[-8+72\ln2\right]-\left[-5\right]\\ &=-8+72\ln2+5\\ &=\boxed{\boxed{72\ln2-3}} \end{aligned}\)

Jadi, \(\displaystyle\intop_5^8\dfrac{f^{-1}(x-1)}{g(3)}\ dx=72\ln2-3\).
JAWAB: A

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No.

Jika tabel berikut menyatakan hasil fungsi f dan g

x	0	1	2	3
f(x)	1	3	1	−1
g(x)	2	0	1	2

maka nilai (f ∘ g ∘ f)(0) + (g ∘ f ∘ g)(1) =

1. 4
2. 3
3. 2

4. 1
5. 0

Grup Matematika Idhamdaz

ALTERNATIF PENYELESAIAN

\begin{aligned} (f\circ g\circ f)(0)+(g\circ f\circ g)(1)&=f(g(f(0)))+g(f(g(1)))\\ &=f(g(1))+g(f(0))\\ &=f(0)+g(1)\\ &=1+0\\ &=\boxed{\boxed{1}} \end{aligned}

Jadi, (f ∘ g ∘ f)(0) + (g ∘ f ∘ g)(1) =1.
JAWAB: D

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No.

Diketahui fungsi f(x) = ax + 6, dengan a ≠ 0 dan \(\left(g\circ f\right)(x) = x + \dfrac6a\), maka nilai dari g(6a) =

1. 10
2. 9
3. 8

4. 7
5. 6

SKMP Juli 2020

ALTERNATIF PENYELESAIAN

\begin{aligned} \left(g\circ f\right)(x) &= x + \dfrac6a\\ g\left(f(x)\right)&=\dfrac{ax}a+ \dfrac6a\\ g\left(ax + 6\right)&=\dfrac{ax+6}a\\ g(x)&=\dfrac{x}a\\ 6(6a)&=\dfrac{6a}a\\ &=\boxed{\boxed{6}} \end{aligned}

Jadi, g(6a) = 6.
JAWAB: E

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