tentukan nilai t yang memenuhi setiap persamaan berikut
a. f^-1 ((t+1)/(t-2)) = 12 dan f(12) = 13
b. f ((1-2t)/(1-2t)) = 7 dan f^-1(7) = -3
a. f^-1 ((t+1)/(t-2)) = 12 dan f(12) = 13
b. f ((1-2t)/(1-2t)) = 7 dan f^-1(7) = -3
Jawaban:
a. t = 9/4
b. t = –1
Pembahasan
Komposisi dan Invers Fungsi
[tex]\large\text{$\boxed{\ \begin{aligned}\\(i)\quad&f(x)=y&\iff&f^{-1}(y)=x\\(ii)\quad&f^{-1}(x)=y&\iff&f(y)=x\\ {}\end{aligned}\ }$}[/tex]
Soal a.
[tex]\large\text{$\begin{aligned}&f^{-1}\left(\frac{t+1}{t-2}\right)=12\:,\ f(12) = 13\\\\{\iff}&\frac{t+1}{t-2}=13\\{\iff}&t+1=13t-26\\{\iff}&t-13t=-26-1\\{\iff}&{-}12t=-27\\{\iff}&4t=9\\{\iff}&t=\bf\frac{9}{4}\end{aligned}$}[/tex]
Soal b.
[tex]\large\text{$\begin{aligned}&f\left(\frac{1-2t}{1+2t}\right)=7\:,\ f^{-1}(7)=-3\\\\{\iff}&\frac{1-2t}{1+2t}=-3\\{\iff}&1-2t=-3-6t\\{\iff}&6t-2t=-3-1\\{\iff}&4t=-4\\{\iff}&t=\bf-1\end{aligned}$}[/tex]
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