The range of which function includes -4?
y = sqrt(x) – 5
y = sqrt(x) + 5
y = sqrt(x + 5)
y = sqrt(x – 5)
The Correct Answer and Explanation is :
The correct answer is: (y = \sqrt{x} – 5)
To determine which function has a range that includes -4, we need to analyze each function individually, focusing on how the square root function behaves and how it affects the overall expression.
- Function: (y = \sqrt{x} – 5)
The square root function, (\sqrt{x}), has a minimum value of 0 (which occurs when (x = 0)). Thus, the minimum value of (y) can be calculated as follows:
[
y_{\text{min}} = 0 – 5 = -5
]
Since (\sqrt{x}) can take values starting from 0 and going to positive infinity, (y) can take values from -5 to positive infinity. Therefore, the range is ([-5, \infty)), which includes -4. - Function: (y = \sqrt{x} + 5)
Here, the minimum value occurs again when (x = 0):
[
y_{\text{min}} = 0 + 5 = 5
]
The range for this function starts at 5 and goes to positive infinity, represented as ([5, \infty)). This range does not include -4. - Function: (y = \sqrt{x + 5})
For this function, the minimum occurs when (x + 5 = 0) (i.e., (x = -5)):
[
y_{\text{min}} = \sqrt{0} = 0
]
Thus, the range is ([0, \infty)), which does not include -4. - Function: (y = \sqrt{x – 5})
The minimum value for this function occurs when (x – 5 = 0) (i.e., (x = 5)):
[
y_{\text{min}} = \sqrt{0} = 0
]
The range is also ([0, \infty)), which does not include -4.
Conclusion:
Among the given functions, the correct answer is (y = \sqrt{x} – 5), as its range is ([-5, \infty)), which includes -4. The other functions do not allow for negative values in their ranges, which disqualifies them from including -4.