Question 1 of 5 Which of the following is the inverse of the following function? \[ f(x)=(x+7)^{2} \] O \( n(x)=-7 \pm \sqrt{x} \) for \( x \leq 0 \) O \( q(x)=7+\sqrt{x} \) for \( x \leq 0 \) \( m(x)=7 \pm \sqrt{x} \) for \( x \geq 0 \) \( 0 \quad p(x)=-7 \pm \sqrt{x} \) for \( x \geq 0 \)

To find the inverse of the function \( f(x) = (x + 7)^2 \), we will follow these steps: 1. **Replace \( f(x) \) with \( y \)**: \[ y = (x + 7)^2 \] 2. **Swap \( x \) and \( y \)**: \[ x = (y + 7)^2 \] 3. **Solve for \( y \)**: - Take the square root of both sides: \[ \sqrt{x} = y + 7 \quad \text{or} \quad -\sqrt{x} = y + 7 \] - Rearranging gives: \[ y = \sqrt{x} - 7 \quad \text{or} \quad y = -\sqrt{x} - 7 \] 4. **Determine the appropriate branch**: Since the original function \( f(x) = (x + 7)^2 \) is defined for all \( x \) but is a parabola opening upwards, we need to restrict the domain to ensure the inverse is a function. The function is increasing for \( x \geq -7 \). Therefore, we will take the positive branch: \[ y = \sqrt{x} - 7 \] 5. **Express the inverse function**: \[ f^{-1}(x) = \sqrt{x} - 7 \] Now, let's compare this with the given options: - \( n(x) = -7 \pm \sqrt{x} \) for \( x \leq 0 \) - \( q(x) = 7 + \sqrt{x} \) for \( x \leq 0 \) - \( m(x) = 7 \pm \sqrt{x} \) for \( x \geq 0 \) - \( p(x) = -7 \pm \sqrt{x} \) for \( x \geq 0 \) The correct inverse function we derived is \( f^{-1}(x) = \sqrt{x} - 7 \), which can be rewritten as: \[ f^{-1}(x) = -7 + \sqrt{x} \quad \text{for } x \geq 0 \] This matches with the option \( p(x) = -7 \pm \sqrt{x} \) for \( x \geq 0 \). Thus, the correct answer is: \[ \boxed{p(x) = -7 \pm \sqrt{x} \text{ for } x \geq 0} \]