Q)Let f:(4,6)-->(6,8) be a function defined by f(x)=x+ [x/2](where [.] denotes the greatest integer function),then f^-1 is equal to

a)x-[x/2] b)-x-2 c)x-2 d)1/(x+[x/2])

Sir,plz solve this .

To find the inverse of the function \( f(x) = x + \left\lfloor \frac{x}{2} \right\rfloor \) defined on the interval \( (4, 6) \) mapping to \( (6, 8) \), we need to analyze the function step by step. The greatest integer function, denoted by \( \left\lfloor x \right\rfloor \), rounds down to the nearest integer. Let's break this down to find \( f^{-1}(x) \).

Understanding the Function

First, let's examine the function \( f(x) \) more closely. The input \( x \) lies within the interval \( (4, 6) \). We can evaluate \( \left\lfloor \frac{x}{2} \right\rfloor \) for this range:

• For \( x \) in \( (4, 6) \), \( \frac{x}{2} \) ranges from \( 2 \) to \( 3 \).
• Thus, \( \left\lfloor \frac{x}{2} \right\rfloor = 2 \) for all \( x \) in this interval.

Now, substituting this back into our function, we get:

\( f(x) = x + 2 \)

Finding the Range of f

Next, we need to determine the output of \( f(x) \) as \( x \) varies from \( 4 \) to \( 6 \):

• When \( x = 4 \), \( f(4) = 4 + 2 = 6 \).
• When \( x = 6 \), \( f(6) = 6 + 2 = 8 \).

Thus, \( f(x) \) maps the interval \( (4, 6) \) to \( (6, 8) \). Now we can express \( f(x) \) as:

\( f(x) = x + 2 \) for \( x \in (4, 6) \).

Finding the Inverse Function

To find the inverse \( f^{-1}(y) \), we need to solve for \( x \) in terms of \( y \). Start with the equation:

\( y = x + 2 \).

Rearranging gives:

\( x = y - 2 \).

Determining the Domain of the Inverse

Since \( f(x) \) maps \( (4, 6) \) to \( (6, 8) \), the inverse function \( f^{-1}(y) \) will take inputs from \( (6, 8) \) and output values in \( (4, 6) \). Therefore, we have:

\( f^{-1}(y) = y - 2 \) for \( y \in (6, 8) \).

Finalizing the Answer

Now, looking at the options provided:

• a) \( x - \left\lfloor \frac{x}{2} \right\rfloor \)
• b) \( -x - 2 \)
• c) \( x - 2 \)
• d) \( \frac{1}{x + \left\lfloor \frac{x}{2} \right\rfloor} \)

The correct expression for the inverse function is \( f^{-1}(x) = x - 2 \), which corresponds to option c). Therefore, the answer is:

c) \( x - 2 \)