Diketahui fungsi g(x) = x - 1 dan (fog) (x) = 4x² - x‚ jika f (b) = 5‚ maka tentukan nilai b! .... Question from @Randomo

Randomo @Randomo

July 2022 1 14 Report

Diketahui fungsi g(x) = x - 1 dan (fog) (x) = 4x² - x‚ jika f (b) = 5‚ maka tentukan nilai b!

henriyulianto

Nilai b adalah ¼ atau –2.

Pembahasan

Komposisi dan Invers Fungsi

Diketahui

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

Ditanyakan

Nilai $b$ jika $f(b)=5$

PENYELESAIAN

Kita tentukan $f(x)$ .

Cara pertama:

$\begin{aligned}4x^2-x&=(f\circ g)(x)\\&=f\big(g(x)\big)\\&=f(x-1)\\\textsf{-----------}&\textsf{------------------------------------------}\\f(x-1)&=4x^2-x\\&=4\left(x^2-2x+1\right)+7x-4\\&=4\left(x^2-2x+1\right)+7x-7+3\\f(x-1)&=4(x-1)^2+7(x-1)+3\\\textsf{-----------}&\textsf{------------------------------------------}\\\therefore\ f(x)&=\boxed{\ 4x^2+7x+3\ }\end{aligned}$

Cara kedua:

$\begin{aligned}f(x)&=f\left(\left(g\circ g^{-1}\right)(x)\right)\\&=\left(f\circ\left(g\circ g^{-1}\right)\right)(x)\\&\quad...\ \textsf{sifat asosiatif}\\&=\left(\left(f\circ g\right)\circ g^{-1}\right)(x)\\&=\left(f\circ g\right)\left(g^{-1}(x)\right)\\&\quad\left[\ \begin{aligned}&g(x)=x-1\\&\Rightarrow g^{-1}(x)=x+1\end{aligned}\right.\\&=\left(f\circ g\right)(x+1)\\&=4(x+1)^2-(x+1)\\&=4\left(x^2+2x+1\right)-x-1\\&=4x^2+8x+4-x-1\\\therefore\ f(x)&=\boxed{\ 4x^2+7x+3\ }\end{aligned}$

Sekarang, kita cari nilai $b$ .

$\begin{aligned}5&=f(b)\\&=4b^2+7b+3\\\Rightarrow 0&=4b^2+7b+3-5\\&=4b^2+7b-2\\\Rightarrow 0&=(4b-1)(b+2)\\\Rightarrow 0&=4b-1\ \;{\rm atau\ \;}0=b+2\\\therefore\ \ &\boxed{\ b=\bf\frac{1}{4}\ }\ \;{\rm atau\ \;}\boxed{\ b=\bf{-}2\ }\end{aligned}$
$\blacksquare$