Find the range of the given function. \[ f(x)=-4-\sqrt{x+3} \] Answer Attempt 1 out of 2 Answer Type: Interval

To understand the range of the function \( f(x) = -4 - \sqrt{x+3} \), we first need to examine the square root term, \( \sqrt{x+3} \). Since the square root outputs only non-negative values, the smallest output occurs when \( x = -3 \), resulting in \( \sqrt{-3+3} = 0 \). Therefore, the largest value of \( f(x) \) can be found at this point: \[ f(-3) = -4 - 0 = -4. \] As \( x \) increases, \( \sqrt{x+3} \) increases, causing \( f(x) \) to decrease. Thus, \( f(x) \) approaches negative infinity. Therefore, the function's range is all values less than or equal to -4: \[ \text{Range: } (-\infty, -4]. \] To truly appreciate the function, it helps to remember that the output changes with the input in unique ways. Consider how \( f(x) \) behaves as \( x \) shifts from -3 and beyond. As \( x \) increases to much larger values, say \( x = 10 \), your output plummets to \( -4 - \sqrt{13} \), which is quite a bit less than -4! This exemplifies the inevitable downward spiral of this fun function.