What is the range of the radical function \( g(x) = -\sqrt{2x+4} + 6 \)?

1. **Determine the domain.** The radicand must be non-negative: \[ 2x + 4 \geq 0. \] Solve for \( x \): \[ 2x \geq -4 \quad \Rightarrow \quad x \geq -2. \] Thus, the domain is \( x \in [-2, \infty) \). 2. **Express the function and evaluate its behavior.** The function is given by: \[ g(x) = -\sqrt{2x+4} + 6. \] Note that \(\sqrt{2x + 4} \geq 0\) on the domain, so the term \(-\sqrt{2x+4}\) is non-positive. This implies: \[ g(x) \leq 6. \] 3. **Determine the maximum value.** The maximum value occurs when \(\sqrt{2x+4}\) is minimized. Since \(\sqrt{2x+4} \geq 0\), its minimum is \(0\), which happens at \(x = -2\). Calculate: \[ g(-2) = -\sqrt{2(-2)+4} + 6 = -\sqrt{0} + 6 = 6. \] 4. **Determine the minimum value.** As \( x \) increases, \(\sqrt{2x+4}\) increases without bound, meaning: \[ g(x) = -\sqrt{2x+4} + 6 \to -\infty \quad \text{as} \quad x \to \infty. \] 5. **State the range.** Combining these results, the function takes all values from negative infinity up to and including \(6\). Therefore, the range of \( g(x) \) is: \[ (-\infty, 6]. \]