Q31 / 90 · JEE Main 2018 (15 Apr Shift 2 Online) · Mathematics

Let $f: \mathrm{A} \rightarrow$ B be a function defined as $f(x)=$ $\frac{x-1}{x-2}$, where $A=R-\{2\}$ and $B=R-\{1\}$. Then $f$ is

  • A
    invertible and $f^{-1}(y)=\frac{2 y+1}{y-1}$
  • B
    invertible and $f^{-1}(y)=\frac{3 y-1}{y-1}$
  • C
    no invertible
  • Correct answer
    D
    invertible and $f^{-1}(y)=\frac{2 y-1}{y-1}$

Solution

Suppose $y=f(x)$ $$ \begin{aligned} &\Rightarrow y=\frac{x-1}{x-2} \\ &\Rightarrow y x-2 y=x-1 \\ &\Rightarrow(y-1) x=2 y-1 \\ &\Rightarrow x=f^{-1}(y)=\frac{2 y-1}{y-1} \end{aligned} $$ As the function is invertible on the given domain and its inverse can be obtained as above.

Hence, the correct answer is option: 'D'.

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