If f\left ( x \right )= \frac{4X+3}{6X-4},x\neq \frac{2}{3}, show that fof\left ( x \right )= x  ofr all x\neq \frac{2}{3}.Also,find the inverse of f.

 

 

 

 
 
 
 
 

Answers (1)

given:       
          f\left ( x \right )= \frac{4x+3}{6x-4}
LHS = fof\left ( x \right )= x\: \: RHS
LHS= fof\left ( x \right )= f\left ( f\left ( x \right ) \right )= \frac{4f\left ( x \right )+3}{6f\left ( x \right )-4}
\Rightarrow \frac{4\left ( \frac{4x+3}{6x-4} \right )+3}{6\left ( \frac{4x+3}{6x-4} \right )-4}\Rightarrow \frac{16x+12+3\left ( 6x-4 \right )}{\left ( 6x-4 \right )\left ( \frac{24x+18-4\left ( 16x-4 \right )}{\not{6x-4}} \right )}
= \frac{16x+18x+\not{12}-\not{12}}{24\not{x}+18-\not{24}x+16}\Rightarrow \frac{3\not{+}x}{34\Rightarrow }x= RHS
\therefore LHS= RHS so f is its own inverse

f(x)=f^{-1}(x)=\frac{4x+3}{6x-4}

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