Find \(f(y)\) if \(f(x)=\frac{x}{x-1}=\frac1y\)

Solution

Given

\[ \frac{x}{x-1}=\frac1y \]

Cross multiply:

\[ xy=x-1 \]

Rearranging,

\[ x-xy=1 \] \[ x(1-y)=1 \] \[ x=\frac1{1-y} \]

Now,

\[ f(y)=\frac{y}{y-1} \]

From

\[ x=\frac1{1-y} \]

we get

\[ 1-y=\frac1x \] \[ y=1-\frac1x \]

Therefore,

\[ f(y)=\frac{1-\frac1x}{\left(1-\frac1x\right)-1} \] \[ =\frac{1-\frac1x}{-\frac1x} \] \[ =-x\left(1-\frac1x\right) \] \[ =-x+1 \] \[ =1-x \]