Question

menentukan invers komposisi fungsi​

Answer 1

Jawab:

a. ⅓ (x - 4)

b. ⅓ (x - 14)

Penjelasan dengan langkah-langkah:

Cara pertama

• Soal a

[tex]\begin{aligned}(f\circ g)(x)&=f(g(x))\\&=f(3x-1)\\&=3x-1+5\\&=3x+4\end{aligned}[/tex]

maka

[tex]\begin{aligned}y&=3x+4\\x&=\frac{y-4}{3}\\(f\circ g)^{-1}(x)&=\frac{x-4}{3}\\\end{aligned}[/tex]

• Soal b

[tex]\begin{aligned}(g\circ f)(x)&=g(f(x))\\&=g(x+5)\\&=3(x+5)-1\\&=3x+14\end{aligned}[/tex]

maka

[tex]\begin{aligned}y&=3x+14\\x&=\frac{y-14}{3}\\(g\circ f)^{-1}(x)&=\frac{x-14}{3}\end{aligned}[/tex]

Cara kedua

• Soal a

[tex]\begin{matrix}\begin{aligned}f(x)&=x+5\\y&=x+5\\x&=y-5\\f^{-1}(x)&=x-5\end{aligned} & \begin{aligned}g(x)&=3x-1\\y&=3x-1\\x&=\frac{y+1}{3}\\g^{-1}(x)&=\frac{x+1}{3}\end{aligned}\end{matrix}[/tex]

maka

[tex]\begin{aligned}(f\circ g)^{-1}(x)&=(g^{-1}\circ f^{-1})(x)\\&=g^{-1}(f^{-1}(x))\\&=g^{-1}(x-5)\\&=\frac{x-5+1}{3}\\&=\frac{x-4}{3}\end{aligned}[/tex]

• Soal b

[tex]\begin{aligned}(g\circ f)^{-1}(x)&=(f^{-1}\circ g^{-1})(x)\\&=f^{-1}(g^{-1}(x))\\&=f^{-1}\left ( \frac{x+1}{3} \right )\\&=\frac{x+1}{3}-5\\&=\frac{x-14}{3}\end{aligned}[/tex]