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

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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:**
   $$
   \boxed{\,[-4,\ \infty)\,}
   $$
The range of the function $$ f(x) = -4 + \sqrt{x + 1} $$ is all real numbers greater than or equal to -4, expressed as $$[-4, \infty)$$.

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

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