If `f(x)=x^(2)+2` then `f^(-1)(x)` is :

Options

A. `sqrt(x-2)`
B. `sqrt(x+2)`
C. `sqrtx+2`
D. `x+4`

Answer

To find the inverse of the function \( f(x) = x^2 + 2 \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
We start by letting:
\[ y = f(x) = x^2 + 2 \]

### Step 2: Solve for \( x \)
Next, we need to express \( x \) in terms of \( y \). Rearranging the equation gives:
\[ y - 2 = x^2 \]
Now, take the square root of both sides:
\[ x = \sqrt{y - 2} \]
Note that we only consider the positive square root because the function \( f(x) = x^2 + 2 \) is defined for \( x \geq 0 \) (assuming we are looking for the principal branch of the inverse).

### Step 3: Express the inverse function
Now, we can express the inverse function \( f^{-1}(y) \):
\[ f^{-1}(y) = \sqrt{y - 2} \]

### Step 4: Replace \( y \) with \( x \)
To find \( f^{-1}(x) \), we replace \( y \) with \( x \):
\[ f^{-1}(x) = \sqrt{x - 2} \]

### Final Answer
Thus, the inverse function is:
\[ f^{-1}(x) = \sqrt{x - 2} \]

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