If the function is defined by , then is:

Explanation: 1. Set up the Equation: Let . To solve for , we first take the logarithm with base 2 on both sides: 2. Form a Quadratic Equation: Expand and rearrange the terms to form a quadratic equation in : 3. Solve for using the Quadratic Formula: Using , where , , and : 4. Determine the Correct Sign: The domain of the original function is given as . This means . If we choose the negative sign: , then will be less than or equal to , which is outside the domain. If we choose the positive sign: , we can satisfy . Therefore, we take the positive square root: 5. Write the Inverse Function: Replace with and with : Correct Option: (b)

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NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

If the function $f: [1, \infty) \rightarrow [1, \infty)$ is defined by $f(x) = 2^{x(x-1)}$ , then $f^{-1}(x)$ is:

Choose the correct answer:

Explanation

1. Set up the Equation:

Let $y = 2^{x(x-1)}$ .

To solve for $x$ , we first take the logarithm with base 2 on both sides:

\log_2 y = \log_2(2^{x(x-1)})

\log_2 y = x(x - 1)

2. Form a Quadratic Equation:

Expand and rearrange the terms to form a quadratic equation in $x$ :

x^2 - x = \log_2 y

x^2 - x - \log_2 y = 0

3. Solve for $x$ using the Quadratic Formula:

Using $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a=1$ , $b=-1$ , and $c=-\log_2 y$ :

x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-\log_2 y)}}{2(1)}

x = \frac{1 \pm \sqrt{1 + 4 \log_2 y}}{2}

4. Determine the Correct Sign:

The domain of the original function $f(x)$ is given as $[1, \infty)$ . This means $x \geq 1$ .

If we choose the negative sign: $x = \frac{1 - \sqrt{1 + 4 \log_2 y}}{2}$ , then $x$ will be less than or equal to $0.5$ , which is outside the domain.
If we choose the positive sign: $x = \frac{1 + \sqrt{1 + 4 \log_2 y}}{2}$ , we can satisfy $x \geq 1$ .

Therefore, we take the positive square root:

x = \frac{1}{2}\{1 + \sqrt{1 + 4 \log_2 y}\}

5. Write the Inverse Function:

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

f^{-1}(x) = \frac{1}{2}\{1 + \sqrt{1 + 4 \log_2 x}\}

Correct Option: (b)