Question

If function f(x) =$\left( {\dfrac{{{\text{x - }}1}}{{{\text{x + 1}}}}} \right)$, which of the following statements is/are correct?
$
  {\text{A}}{\text{. f}}\left( {\dfrac{1}{{\text{x}}}} \right) = {\text{f}}\left( {\text{x}} \right) \\
  {\text{B}}{\text{. f}}\left( {\dfrac{1}{{\text{x}}}} \right) = - {\text{f}}\left( {\text{x}} \right) \\
  {\text{C}}{\text{. f}}\left( { - \dfrac{1}{{\text{x}}}} \right) = \dfrac{1}{{{\text{f}}\left( {\text{x}} \right)}} \\
  {\text{D}}{\text{. f}}\left( { - \dfrac{1}{{\text{x}}}} \right) = - \dfrac{1}{{{\text{f}}\left( {\text{x}} \right)}} \\
$

Answer 1

Hint: To solve the question, we consider the given equation f(x) in the question and substitute $\dfrac{1}{{\text{x}}}$ and $ - \dfrac{1}{{\text{x}}}$ in place of x respectively and check the options to determine the answer.

Complete Step-by-Step solution:
Given Data,
f (x) =$\left( {\dfrac{{{\text{x - }}1}}{{{\text{x + 1}}}}} \right)$.
Now we substitute $\dfrac{1}{{\text{x}}}$ in place of x for the given equation f(x).
Therefore f ($\dfrac{1}{{\text{x}}}$) = $\left( {\dfrac{{\dfrac{1}{{\text{x}}} - 1}}{{\dfrac{1}{{\text{x}}} + 1}}} \right) = \left( {\dfrac{{1 - {\text{x}}}}{{1 + {\text{x}}}}} \right) = - {\text{f}}\left( {\text{x}} \right)$
Hence Option B is verified.
Now we substitute -$\dfrac{1}{{\text{x}}}$ in place of x for the given equation f(x).
And f ($ - \dfrac{1}{{\text{x}}}$) = $\left( {\dfrac{{ - \dfrac{1}{{\text{x}}} - 1}}{{ - \dfrac{1}{{\text{x}}} + 1}}} \right) = \left( {\dfrac{{ - 1 - {\text{x}}}}{{ - 1 + {\text{x}}}}} \right) = \left( {\dfrac{{1 + {\text{x}}}}{{1 - {\text{x}}}}} \right) = - \left( {\dfrac{{1 + {\text{x}}}}{{{\text{x - 1}}}}} \right) = - \dfrac{1}{{{\text{f}}\left( {\text{x}} \right)}}$
Hence Option D is verified.
Hence, Options B and D are correct.

Note: In order to solve this type of questions where we are asked to verify options and determine the answer the key is to substitute the value of (x) given in the options in the given equation f(x) one by one, then solve. Then we pick out whichever option holds right as the correct answer.