Exercise Zone : Invers Fungsi
Table of Contents
Tipe:
No.
Tentukan f−1(x) dari \({f(x)=\dfrac{3x+5}{2x-3}}\)ALTERNATIF PENYELESAIAN
\begin{aligned}
f(x)&=\dfrac{3x+5}{2x-3}\\[3.5pt]
y&=\dfrac{3x+5}{2x-3}\\[3.5pt]
(2x-3)y&=3x+5\\
2xy-3y&=3x+5\\
2xy-3x&=3y+5\\
x(2y-3)&=3y+5\\
x&=\dfrac{3y+5}{2y-3}\\
f^{-1}(x)&=\boxed{\boxed{\dfrac{3x+5}{2x-3}}}
\end{aligned}
Jadi, \(f^{-1}(x)=\dfrac{3x+5}{2x-3}\).
No.
Diketahui fungsi f dengan rumus f(x) = 3x + 1 dan f−1(x) adalah fungsi invers dari f(x). Nilai dari f−1(7) = ....- 1
- 2
- 3
- 4
- 11
ALTERNATIF PENYELESAIAN
Misal f−1(7) = x
\begin{aligned}
f(x)&=7\\
3x+1&=7\\
3x&=6\\
x&=2\\
f^{-1}(7)&=\boxed{\boxed{2}}
\end{aligned}
Jadi, f−1(7) = 2.
JAWAB: B
JAWAB: B
No.
fungsi invers f−1(x) dari f(x) = 23x adalahALTERNATIF PENYELESAIAN
\begin{aligned}
y&=2^{3x}\\
3x&={^2\negmedspace\log y}\\
x&=\dfrac13\ {^2\negmedspace\log y}\\
f^{-1}(x)&=\dfrac13\ {^2\negmedspace\log x}
\end{aligned}
Jadi, \(f^{-1}(x)=\dfrac13\ {^2\negmedspace\log x}\).
No.
Tentukan rumus fungsi invers untuk fungsi \(f(x) = \dfrac{2x+5}{3x-1}\)ALTERNATIF PENYELESAIAN
\begin{aligned}y&=\dfrac{2x+5}{3x-1}\\(3x-1)y&=2x+5\\3xy-y&=2x+5\\3xy-2x&=y+5\\x(3y-2)&=y+5\\x&=\dfrac{y+5}{3y-2}\\f^{-1}(x)&=\boxed{\boxed{\dfrac{x+5}{3x-2}}}\end{aligned}
Jadi, rumus fungsi invers untuk fungsi \(f(x) = \dfrac{2x+5}{3x-1}\) adalah \(f^{-1}(x)=\dfrac{x+5}{3x-2}\).
No.
DiketahuiALTERNATIF PENYELESAIAN
\begin{aligned}\left(f\circ g\right)(x)&=f\left(g(x)\right)\\&=f(3x-2)\\&=\dfrac{3x-2-4}{3x-2+3}\\y&=\dfrac{3x-6}{3x+1}\\(3x+1)y&=3x-6\\3xy+y&=3x-6\\3xy-3x&=-y-6\\x(3y-3)&=-y-6\\x&=\dfrac{-y-6}{3y-3}\\(f\circ g)^{-1}(x)&=\dfrac{-x-6}{3x-3}\end{aligned}
Jadi, \(\left(f\circ g\right)^{-1} (x)=\dfrac{-x-6}{3x-3}\).
No.
Diketahui \(f(x)= \dfrac{3x+1}{x-5}\),ALTERNATIF PENYELESAIAN
\begin{aligned}f(x)&=\dfrac{3x+1}{x-5}\\y&=\dfrac{3x+1}{x-5}\\(x-5)y&=3x+1\\xy-5y&=3x+1\\xy-3x&=5y+1\\x(y-3)&=5y+1\\x&=\dfrac{5y+1}{y-3}\\f^{-1}(x)&=\dfrac{5x+1}{x-3}\\f^{-1}(11)&=\dfrac{5(11)+1}{11-3}\\&=\dfrac{55+1}8\\&=\dfrac{56}8\\&=\boxed{\boxed{7}}\end{aligned}
Jadi, f−1(11) = 7.
No.
DiketahuiALTERNATIF PENYELESAIAN
$\begin{aligned}
(f\circ g)(x)&=f(g(x))\\
&=f(5a-2)\\
&=12(5a-2)+4\\
&=60a-24+4\\
&=60a-20
\end{aligned}
\begin{aligned}
\left(g^{-1}\circ f^{-1}\right)(x)&=(f\circ g)^{-1}(x)\\
&=\boxed{\boxed{\dfrac{a+20}{60}}}
\end{aligned}
Jadi, \(\left(g^{-1}\circ f^{-1}\right)(a)=\dfrac{a+20}{60}\).
No.
JikaALTERNATIF PENYELESAIAN
Misal k−1(64) = a, maka
\(\begin{aligned} k(a)&=64\\ 2^{a+1}&=2^6\\ a+1&=6\\ a&=5\\ k^{-1}(64)&=\color{blue}\boxed{\boxed{\color{black}5}} \end{aligned}\)
\(\begin{aligned} k(a)&=64\\ 2^{a+1}&=2^6\\ a+1&=6\\ a&=5\\ k^{-1}(64)&=\color{blue}\boxed{\boxed{\color{black}5}} \end{aligned}\)
Jadi, k−1(64) = 5.
No.
Diketahui- 5
- 4
- 3
- 2
- 1
ALTERNATIF PENYELESAIAN
Misal f −1(−1) = p , maka
\(\begin{aligned} f(a)&=-1\\ 4a-5&=-1\\ 4a&=4\\ a&=1\\ f^{-1}(-1)&=\color{blue}\boxed{\boxed{\color{black}1}} \end{aligned}\)
\(\begin{aligned} f(a)&=-1\\ 4a-5&=-1\\ 4a&=4\\ a&=1\\ f^{-1}(-1)&=\color{blue}\boxed{\boxed{\color{black}1}} \end{aligned}\)
Jadi, f −1(−1) = 1.
JAWAB: E
JAWAB: E
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