Question

If $f\left(x\right)=\frac{x}{x-1}=\frac{1}{y},\mathrm{then}f\left(y\right)=$ __________ .

Answer 1

f(x) = $\frac{x}{x-1}$
given $\frac{x}{x-1}=\frac{1}{y}$
xy = x − 1 i.e y = $\frac{x-1}{x}$
$\begin{array}{rcl}\mathrm{i}.\mathrm{e}f\left(y\right)& =& \frac{y}{y-1}\\ & =& \frac{{\displaystyle \frac{x-1}{x}}}{{\displaystyle \frac{x-1}{x}}-1}\\ & =& \frac{{\displaystyle \frac{x-1}{x}}}{{\displaystyle \frac{x-1-x}{x}}}\\ & =& \frac{x-1}{x}\times \frac{x}{-1}\\ & =& -\left(x-1\right)\\ \therefore f\left(y\right)& =& 1-x\end{array}$