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

Options

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

Answer

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

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = x^2 + 2 \]
We can rewrite this as:
\[ y = x^2 + 2 \]

### Step 2: Solve for \( x \)
Next, we need to express \( x \) in terms of \( y \). We rearrange the equation:
\[ y - 2 = x^2 \]
Now, we take the square root of both sides to solve for \( x \):
\[ x = \sqrt{y - 2} \]
Note that we consider only the positive root because the function \( f(x) = x^2 + 2 \) is not one-to-one unless we restrict the domain to non-negative \( x \).

### Step 3: Write 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 express the inverse function in standard form, we replace \( y \) with \( x \):
\[ f^{-1}(x) = \sqrt{x - 2} \]

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

---