If `f:[1, infty) rarr [2, infty]` is given by `f(x)=x+1/x," find " f^(-1)(x)` (assume bijection).

Answer

To find the inverse of the function \( f(x) = x + \frac{1}{x} \) defined on the interval \([1, \infty)\) mapping to \([2, \infty)\), we will follow these steps:

### Step 1: Set up the equation
We start by setting \( f(x) = y \):
\[ y = x + \frac{1}{x} \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ x + \frac{1}{x} - y = 0 \]
This can be rewritten as:
\[ x^2 - yx + 1 = 0 \]

### Step 3: Identify the quadratic formula
This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = -y \)
- \( c = 1 \)

### Step 4: Apply the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-(-y) \pm \sqrt{(-y)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{y \pm \sqrt{y^2 - 4}}{2} \]

### Step 5: Determine the correct sign
Since \( f(x) \) is defined for \( x \geq 1 \) and we are looking for positive values, we only consider the positive root:
\[ x = \frac{y + \sqrt{y^2 - 4}}{2} \]

### Step 6: Express the inverse function
Thus, we can express the inverse function \( f^{-1}(y) \) as:
\[ f^{-1}(y) = \frac{y + \sqrt{y^2 - 4}}{2} \]

### Step 7: Replace \( y \) with \( x \)
Finally, to express \( f^{-1}(x) \):
\[ f^{-1}(x) = \frac{x + \sqrt{x^2 - 4}}{2} \]

### Final Answer
The inverse function is:
\[ f^{-1}(x) = \frac{x + \sqrt{x^2 - 4}}{2}, \quad x \geq 2 \]

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