NIMCET 2022 — Mathematics PYQ

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Question

NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

Inverse of the function $f (x) = \frac{1 0 ^{x} - 1 0 ^{- x}}{1 0 ^{x} + 1 0 ^{- x}}$ is

Choose the correct answer:

Explanation

To find the inverse of the function, we follow the standard procedure of replacing $f(x)$ with $y$ and then solving for $x$ in terms of $y$ .

1. Set up the equation:

Let $y = \frac{10^x - 10^{-x}}{10^x + 10^{-x}}$

2. Use Componendo and Dividendo:

To simplify the fraction, we apply the rule: if $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a+b}{a-b} = \frac{c+d}{c-d}$ .

\frac{y + 1}{y - 1} = \frac{(10^x - 10^{-x}) + (10^x + 10^{-x})}{(10^x - 10^{-x}) - (10^x + 10^{-x})}

\frac{1 + y}{-(1 - y)} = \frac{2 \cdot 10^x}{-2 \cdot 10^{-x}}

\frac{1 + y}{1 - y} = \frac{10^x}{10^{-x}}

3. Simplify the powers:

\frac{1 + y}{1 - y} = 10^x \cdot 10^x

\frac{1 + y}{1 - y} = 10^{2x}

4. Take Logarithm (base 10) on both sides:

\log_{10} \left( \frac{1 + y}{1 - y} \right) = \log_{10}(10^{2x})

\log_{10} \left( \frac{1 + y}{1 - y} \right) = 2x

5. Isolate $x$ :

x = \frac{1}{2} \log_{10} \left( \frac{1 + y}{1 - y} \right)

6. Final Step:

Replace $x$ with $f^{-1}(x)$ and $y$ with $x$ :

f^{-1}(x) = \frac{1}{2} \log_{10} \left( \frac{1 + x}{1 - x} \right)

Conclusion:

The inverse of the given function matches option B.

Correct Option:

B) $\frac{1}{2} \log_{10} \left( \frac{1 + x}{1 - x} \right)$