Correct Answer - Option 1 :
\(\dfrac{3f(x)}{2f(x)+1}\)
Concept:
According to the rule of a linear function, we know that, if, \(f(x)=x\),
\(\Rightarrow f(ax)=ax\)
where a is the coefficient of x.
Calculation:
\(f(x)=\dfrac{x}{x-1}\) ---(1)
\(\Rightarrow(x-1)f(x)=x\)
\(\Rightarrow xf(x)-f(x)=x\)
\(\Rightarrow xf(x)-x=f(x)\)
\(\Rightarrow x(f(x)-1)=f(x)\)
\(\Rightarrow x=\dfrac{f(x)}{f(x)-1}\) ---(2)
Replacing x with 3x in equation(1), we get
\(f(3x)=\dfrac{3x}{3x-1}\)
Now, substituting the value of x from equation(2) in this equation, we get
\(f(3x)=\dfrac{\dfrac{3f(x)}{f(x)-1}}{\dfrac{3f(x)}{f(x)-1}-1}\)
\(\Rightarrow f(3x)=\dfrac{\dfrac{3f(x)}{f(x)-1}}{\dfrac{3f(x)-f(x)+1}{f(x)-1}}\)
\(\Rightarrow f(3x)=\dfrac{3f(x)}{2f(x)+1}\)
Hence, \(f(3x)=\dfrac{3f(x)}{2f(x)+1}\)
