f (x) is any function and `f^(-1)(x)` is known as inverse of `f (x)`, then `f^(-1)(x)` of `f(x)=(1)/(x)+1` is

Options

A. `(1)/(x)-1`
B. `x-1`
C. `(1)/((x-1))`
D. `(1)/(x+1)`

Answer

To find the inverse function \( f^{-1}(x) \) of the function \( f(x) = \frac{1}{x} + 1 \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
Let \( f(x) = y \):
\[ y = \frac{1}{x} + 1 \]

### Step 2: Solve for \( x \) in terms of \( y \)
To find the inverse, we need to express \( x \) in terms of \( y \). Start by isolating \( \frac{1}{x} \):
\[ y - 1 = \frac{1}{x} \]

Now, take the reciprocal of both sides:
\[ x = \frac{1}{y - 1} \]

### Step 3: Write the inverse function
Since we have expressed \( x \) in terms of \( y \), we can write the inverse function:
\[ f^{-1}(y) = \frac{1}{y - 1} \]

### Step 4: Replace \( y \) with \( x \)
To express the inverse function in standard notation, replace \( y \) with \( x \):
\[ f^{-1}(x) = \frac{1}{x - 1} \]

### Final Answer
Thus, the inverse of the function \( f(x) = \frac{1}{x} + 1 \) is:
\[ f^{-1}(x) = \frac{1}{x - 1} \]

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