Question

Establish which of the following options is the inverse of the function f(x)=x2−6
with a domain restricted to x≥0
.(1 point)
Responses

f−1(x)=x−−√+6
f inverse left parenthesis x right parenthesis equals Start Root x End Root plus 6

f−1(x)=x+6−−−−−√
f inverse left parenthesis x right parenthesis equals Start Root x plus 6 End Root

f−1(x)=x−6−−−−−√
f inverse left parenthesis x right parenthesis equals Start Root x minus 6 End Root

f−1(x)=x+6–√

Answer 1

To find the inverse of the function \( f(x) = x^2 - 6 \) with the domain restricted to \( x \geq 0 \), we need to follow these steps:

1. Replace \( f(x) \) with \( y \):
\[ y = x^2 - 6 \]

2. Solve for \( x \) in terms of \( y \):
\[ y + 6 = x^2 \]
\[ x^2 = y + 6 \]
\[ x = \sqrt{y + 6} \] (Since \( x \geq 0 \))

3. Therefore, we find that:
\[ f^{-1}(y) = \sqrt{y + 6} \]

4. Replace \( y \) back with \( x \) to express the inverse function:
\[ f^{-1}(x) = \sqrt{x + 6} \]

Thus, the correct inverse function is:
**\( f^{-1}(x) = \sqrt{x + 6} \)**

From the given responses, the correct option is:
- \( f^{-1}(x) = \sqrt{x + 6} \) (This is the same as the first option listed in your response).