Question
Identify the square root function that is increasing over the interval [−3,∞).(1 point)
Responses
y=x−3−−−−−√+2
y is equal to square root of x minus 3 end root plus 2
y=−x+3−−−−−√
y is equal to negative square root of x plus 3 end root
y=x+3−−−−−√+2
y is equal to square root of x plus 3 end root plus 2
y=−x−3−−−−−−√+2
Responses
y=x−3−−−−−√+2
y is equal to square root of x minus 3 end root plus 2
y=−x+3−−−−−√
y is equal to negative square root of x plus 3 end root
y=x+3−−−−−√+2
y is equal to square root of x plus 3 end root plus 2
y=−x−3−−−−−−√+2
Answers
GPT-4o mini
To identify the square root function that is increasing over the interval \([-3, \infty)\), let's analyze the given options.
1. **\(y = \sqrt{x - 3} + 2\)**
- The expression inside the square root, \(x - 3\), requires \(x \geq 3\) for \(y\) to be defined. Therefore, this function is not defined on \([-3, 3)\) and starts from \(x = 3\).
2. **\(y = -\sqrt{x + 3} + 2\)**
- The expression inside the square root, \(x + 3\), requires \(x \geq -3\). However, since there's a negative sign in front of the square root, this function is decreasing in the interval \([-3, \infty)\).
3. **\(y = \sqrt{x + 3} + 2\)**
- This function is defined for \(x \geq -3\), and since it is a square root function with a positive leading coefficient, it is increasing over the entire interval \([-3, \infty)\).
4. **\(y = -\sqrt{-x - 3} + 2\)**
- The expression inside the square root, \(-x - 3\), requires \(x \leq -3\) for \(y\) to be defined. And again, with the negative sign in front, this function is decreasing on the defined interval.
From this analysis, the function that is increasing over the interval \([-3, \infty)\) is:
**\(y = \sqrt{x + 3} + 2\)**
1. **\(y = \sqrt{x - 3} + 2\)**
- The expression inside the square root, \(x - 3\), requires \(x \geq 3\) for \(y\) to be defined. Therefore, this function is not defined on \([-3, 3)\) and starts from \(x = 3\).
2. **\(y = -\sqrt{x + 3} + 2\)**
- The expression inside the square root, \(x + 3\), requires \(x \geq -3\). However, since there's a negative sign in front of the square root, this function is decreasing in the interval \([-3, \infty)\).
3. **\(y = \sqrt{x + 3} + 2\)**
- This function is defined for \(x \geq -3\), and since it is a square root function with a positive leading coefficient, it is increasing over the entire interval \([-3, \infty)\).
4. **\(y = -\sqrt{-x - 3} + 2\)**
- The expression inside the square root, \(-x - 3\), requires \(x \leq -3\) for \(y\) to be defined. And again, with the negative sign in front, this function is decreasing on the defined interval.
From this analysis, the function that is increasing over the interval \([-3, \infty)\) is:
**\(y = \sqrt{x + 3} + 2\)**