The range of which function includes -4?

[tex]y=\sqrt{x}-5[/tex]

[tex]y=\sqrt{x}+5[/tex]

[tex]y=\sqrt{x+5}[/tex]

[tex]y=\sqrt{x-5}[/tex]

Analyze each function's range based on the properties of the square root function.
For y = x ​ − 5 , the range is y ≥ − 5 , which includes -4.
For y = x ​ + 5 , the range is y ≥ 5 , which does not include -4.
For y = x + 5 ​ and y = x − 5 ​ , the range is y ≥ 0 , which does not include -4. Therefore, the answer is y = x ​ − 5 . y = x ​ − 5 ​

Explanation

Analyzing the Functions We are given four functions and we want to determine which of them has a range that includes -4. Let's analyze each function separately.

Analyzing the first function

y = x ​ − 5 : Since the square root function x ​ always returns a non-negative value (i.e., x ​ ≥ 0 ), the smallest possible value for y is when x ​ = 0 , which gives y = 0 − 5 = − 5 . Thus, the range of this function is y ≥ − 5 . Since − 4 ≥ − 5 , -4 is in the range of this function.

Analyzing the second function

y = x ​ + 5 : Similarly, since x ​ ≥ 0 , the smallest possible value for y is when x ​ = 0 , which gives y = 0 + 5 = 5 . Thus, the range of this function is y ≥ 5 . Since − 4 < 5 , -4 is not in the range of this function.

Analyzing the third function

y = x + 5 ​ : The domain of this function is x + 5 ≥ 0 , which means x ≥ − 5 . Since x + 5 ​ ≥ 0 , the smallest possible value for y is when x + 5 ​ = 0 , which gives y = 0 . Thus, the range of this function is y ≥ 0 . Since − 4 < 0 , -4 is not in the range of this function.

Analyzing the fourth function

y = x − 5 ​ : The domain of this function is x − 5 ≥ 0 , which means x ≥ 5 . Since x − 5 ​ ≥ 0 , the smallest possible value for y is when x − 5 ​ = 0 , which gives y = 0 . Thus, the range of this function is y ≥ 0 . Since − 4 < 0 , -4 is not in the range of this function.

Final Answer Therefore, the range of the function y = x ​ − 5 includes -4.


Examples
Understanding function ranges is crucial in many real-world applications. For example, if you're designing a sensor that measures temperature, you need to know the range of temperatures the sensor can accurately detect. If the sensor's output is modeled by a function, knowing the range helps you determine if the sensor can measure temperatures down to -4 degrees Celsius, which might be important in certain environments.

Answered by GinnyAnswer | 2025-07-03

The function that includes -4 in its range is y = x ​ − 5 since its range is y ≥ − 5 . Other functions do not include -4 in their ranges as shown in the analysis. Therefore, the answer is y = x ​ − 5 .
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Answered by Anonymous | 2025-07-04