Find the range of the given function. \[ f(x)=-4+\sqrt{x+1} \] Answer Attempt 1 out of 2 Answer Type: Interval

To find the **range** of the function \( f(x) = -4 + \sqrt{x + 1} \), follow these steps: 1. **Determine the Domain:** - The expression inside the square root must be non-negative: \[ x + 1 \geq 0 \implies x \geq -1 \] - **Domain:** \( x \geq -1 \) 2. **Analyze the Function:** - The square root function \( \sqrt{x + 1} \) yields values: \[ \sqrt{x + 1} \geq 0 \] - Therefore, the smallest value of \( \sqrt{x + 1} \) is 0 (when \( x = -1 \)), and it increases without bound as \( x \) increases. 3. **Compute the Range:** - Subtracting 4 from the square root: \[ f(x) = -4 + \sqrt{x + 1} \geq -4 + 0 = -4 \] - As \( \sqrt{x + 1} \) increases, \( f(x) \) also increases without any upper limit. - **Range:** All real numbers \( f(x) \) such that \( f(x) \geq -4 \) 4. **Express in Interval Notation:** \[ \boxed{\,[-4,\ \infty)\,} \]