Question
Y= 4 3 sqaure root x - 2
Select the description that corresponds to the graphed function. (1 point) Responses The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers. The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers. The domain and range of the function are both the set of all real numbers. The domain and range of the function are both the set of all real numbers. The domain of the function is the set of all real numbers and the range is the set of all non-negative real numbers. The domain of the function is the set of all real numbers and the range is the set of all non-negative real numbers. The domain and range of the function are both the set of all non-negative real numbers.
Select the description that corresponds to the graphed function. (1 point) Responses The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers. The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers. The domain and range of the function are both the set of all real numbers. The domain and range of the function are both the set of all real numbers. The domain of the function is the set of all real numbers and the range is the set of all non-negative real numbers. The domain of the function is the set of all real numbers and the range is the set of all non-negative real numbers. The domain and range of the function are both the set of all non-negative real numbers.
Answers
GPT-4o mini
To analyze the function \( Y = 4 \sqrt{x} - 2 \), we will determine the domain and range.
**Domain:**
The term \( \sqrt{x} \) implies that \( x \) must be non-negative for the function to be defined. Therefore, the domain is:
\[
x \geq 0 \quad \text{or} \quad [0, \infty)
\]
This means the domain of the function is the set of all non-negative real numbers.
**Range:**
To find the range, we can evaluate \( Y \):
- Since \( \sqrt{x} \) can take on any non-negative value, \( 4 \sqrt{x} \) also takes on all non-negative values but scaled by 4. Thus, as \( x \) approaches 0, \( Y \) approaches \( 4 \cdot 0 - 2 = -2 \).
- As \( x \) increases to infinity, \( 4 \sqrt{x} \) increases to infinity as well, which means \( Y \) also increases to infinity.
Therefore, the minimum value of \( Y \) is \(-2\) (when \( x = 0\)), and there is no upper bound, resulting in the range:
\[
Y \geq -2 \quad \text{or} \quad [-2, \infty)
\]
Based on this analysis, we conclude that:
- The domain is the set of all non-negative real numbers.
- The range is \([-2, \infty)\), which is not one of the provided options.
However, among the options given, the one that most closely aligns with our findings is:
**The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers.**
Please note that while this exact range is slightly off, it characterizes that the function is defined for \(x \geq 0\).
Thus, the most accurate choice is:
**"The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers."**
**Domain:**
The term \( \sqrt{x} \) implies that \( x \) must be non-negative for the function to be defined. Therefore, the domain is:
\[
x \geq 0 \quad \text{or} \quad [0, \infty)
\]
This means the domain of the function is the set of all non-negative real numbers.
**Range:**
To find the range, we can evaluate \( Y \):
- Since \( \sqrt{x} \) can take on any non-negative value, \( 4 \sqrt{x} \) also takes on all non-negative values but scaled by 4. Thus, as \( x \) approaches 0, \( Y \) approaches \( 4 \cdot 0 - 2 = -2 \).
- As \( x \) increases to infinity, \( 4 \sqrt{x} \) increases to infinity as well, which means \( Y \) also increases to infinity.
Therefore, the minimum value of \( Y \) is \(-2\) (when \( x = 0\)), and there is no upper bound, resulting in the range:
\[
Y \geq -2 \quad \text{or} \quad [-2, \infty)
\]
Based on this analysis, we conclude that:
- The domain is the set of all non-negative real numbers.
- The range is \([-2, \infty)\), which is not one of the provided options.
However, among the options given, the one that most closely aligns with our findings is:
**The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers.**
Please note that while this exact range is slightly off, it characterizes that the function is defined for \(x \geq 0\).
Thus, the most accurate choice is:
**"The domain of the function is the set of all non-negative real numbers and the range is the set of all real numbers."**