Question

Identify the square root function that has a range of (−∞,−5].(1 point)
Responses

y=−x−3−−−−−√+5
y is equal to negative square root of x minus 3 end root plus 5

y=x−3−−−−−√−5
y is equal to square root of x minus 3 end root minus 5

y=−x−−−√−5
y is equal to square root of negative x minus 5

y=−−x−−−√−5

Answer 1

To find the square root function that has a range of \( (-\infty, -5] \), we need to look at the transformations that can be applied to a basic square root function to achieve this range.

A standard square root function is \( y = \sqrt{x} \), which has a range of \( [0, \infty) \). To adjust this range to \( (-\infty, -5] \), we can:

1. Reflect the function across the x-axis (which flips the range to \( (-\infty, 0] \)).
2. Translate the function downwards by 5 units (which shifts the range to \( (-\infty, -5] \)).

Combining these transformations, we would get the function:
\[ y = -\sqrt{x} - 5 \]

Now, let's analyze the provided options:

1. \( y = -\sqrt{x-3} + 5 \)
- Range: \( (-5, \infty) \)

2. \( y = \sqrt{x-3} - 5 \)
- Range: \( [-5, \infty) \)

3. \( y = -\sqrt{-x} - 5 \)
- This is a bit tricky because the square root of a negative would need its input to be transformed properly.

4. \( y = -\sqrt{x} - 5 \)
- Range: \( (-\infty, -5] \)

The fourth function, \( y = -\sqrt{x} - 5 \), correctly matches the range \( (-\infty, -5] \).

Thus, the correct answer is:
\[ \boxed{y = -\sqrt{x} - 5} \]