Let f
Question:

Let $\boldsymbol{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $\boldsymbol{f}(\mathrm{x})=2 \mathrm{x}-1$ and $\mathrm{g}: \mathrm{R}-\{1\} \rightarrow \mathrm{R}$ be defined as $\mathrm{g}(\mathrm{x})=\frac{\mathrm{x}-\frac{1}{2}}{\mathrm{x}-1}$. Then the composition function $f(g(x))$ is :

  1. (1) both one-one and onto

  2. (2) onto but not one-one

  3. (3) neither one-one nor onto

  4. (4) one-one but not onto


Correct Option: , 4

Solution:

$f(g(x))=2 g(x)-1$

$=2 \frac{\left(\frac{x-\frac{1}{2}}{2}\right)}{x-1}=\frac{x}{x-1}$

$f(g(x))=1+\frac{1}{x-1}$

one-one, into

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