If \(f(c)=\frac{3}{2c-3}\), find \( \frac{kn^2}{lm}\) when \(f^{-1}(c)\times c \times f(c)\) equals the simplified fraction\(\frac{kc+l}{mc+n}\), where \(k,l,m,\text{ and }n\) are integers.

Guest Jul 22, 2020

#3**+1 **

**If**

**\(f(c)=\dfrac{3}{2c-3}\), ****find **\(\dfrac{kn^2}{lm} \)

**when \(f^{-1}(c)\times c \times f(c)\) equals the simplified fraction \(\dfrac{kc+l}{mc+n}\), where \(k\), \(l\), \(m\), and \(n\) are integers.**

**My attempt:**

\(\begin{array}{|rcll|} \hline f(c) &=& \dfrac{3}{2c-3} \quad | \quad c=f^{-1}(c) \\\\ f\Big(f^{-1}(c)\Big) &=& \dfrac{3}{2f^{-1}(c)-3} \quad | \quad f\Big(f^{-1}(c)\Big) = c \\\\ c &=& \dfrac{3}{2f^{-1}(c)-3} \\\\ c\Big(2f^{-1}(c)-3\Big) &=& 3 \\\\ 2cf^{-1}(c)-3c &=& 3 \\\\ 2cf^{-1}(c) &=& 3c+3 \\\\ cf^{-1}(c) &=& \dfrac{3c+3}{2} \\\\ \mathbf{f^{-1}(c)\times c} &=& \mathbf{\dfrac{3c+3}{2}} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline f^{-1}(c)\times c \times f(c) &=& \dfrac{(3c+3)}{2}\times \dfrac{3}{(2c-3)} \\\\ f^{-1}(c)\times c \times f(c) &=& \dfrac{3(3c+3)}{2(2c-3)} \\\\ \mathbf{f^{-1}(c)\times c \times f(c)} &=& \mathbf{\dfrac{9c+9}{4c-6}} \\ \hline && \boxed{k=9,\ l=9,\ m=4,\ n=-6} \\\\ \dfrac{kn^2}{lm} &=& \dfrac{9(-6)^2}{9*4} \\\\ \dfrac{kn^2}{lm} &=& \dfrac{36}{4} \\\\ \mathbf{\dfrac{kn^2}{lm}} &=& \mathbf{9} \\ \hline \end{array}\)

heureka Jul 23, 2020