Let f
Question:

Let $f: \mathrm{R}-\{3\} \rightarrow \mathrm{R}-\{1\}$ be defined by $f(\mathrm{x})=\frac{\mathrm{x}-2}{\mathrm{x}-3}$. Let $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be given as $g(x)=2 x-3$. Then, the sum of all the values of $x$ for which $f^{-1}(x)+g-1(x)=\frac{13}{2}$ is equal to

  1. (1) 7

  2. (2) 2

  3. (3) 5

  4. (4) 3


Correct Option: 3,

Solution:

$f(x)=y=\frac{x-2}{x-3}$

$\therefore x=\frac{3 y-2}{y-1}$

$\therefore f^{-1}(x)=\frac{3 x-2}{x-1}$

$\backslash \& g(x)=y=2 x-3$

$\therefore x=\frac{y+3}{2}$

$\therefore g^{-1}(x)=\frac{x+3}{2}$

$\because f^{-1}(x)+g^{-1}(x)=\frac{13}{2}$

$\therefore$ sum of roots

$x_{1}+x_{2}=5$

 

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