Question

Establish which of the following options is the inverse of the function f(x)=x^2−6
with a domain restricted to x≥0
.(1 point)
Responses

Answer 1

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

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

2. **Swap \( x \) and \( y \)**:
\[
x = y^2 - 6
\]

3. **Solve for \( y \)**:
\[
y^2 = x + 6
\]
\[
y = \sqrt{x + 6}
\]
(Since the domain is restricted to \( x \geq 0 \), we will only take the positive square root.)

4. **Write the inverse function**:
\[
f^{-1}(x) = \sqrt{x + 6}
\]

Therefore, the inverse of the function \( f(x) = x^2 - 6 \) with a domain restricted to \( x \geq 0 \) is:

\[
f^{-1}(x) = \sqrt{x + 6}
\]

If you have multiple-choice options, look for this expression as the correct inverse.