Question
The graph of g(x) is shown.
On a coordinate plane, a piecewise function has 2 lines. The first line has a closed circle at (negative 4, 4) and goes down to a closed circle at (negative 1, 1). The second line has a closed circle at (0, 1) and goes do to an open circle at (3, negative 1).
Which statements describe the domain and range of g(x)? Select two options.
The function g(x) is defined for all real numbers x.
The maximum value of the range is 4.
The maximum value of the domain is 3.
The range of g(x) is {y| –1 < y ≤ 4}.
The domain of g(x) is {x| –4 < x ≤ 3}.
On a coordinate plane, a piecewise function has 2 lines. The first line has a closed circle at (negative 4, 4) and goes down to a closed circle at (negative 1, 1). The second line has a closed circle at (0, 1) and goes do to an open circle at (3, negative 1).
Which statements describe the domain and range of g(x)? Select two options.
The function g(x) is defined for all real numbers x.
The maximum value of the range is 4.
The maximum value of the domain is 3.
The range of g(x) is {y| –1 < y ≤ 4}.
The domain of g(x) is {x| –4 < x ≤ 3}.
Answers
GPT-4o mini
To analyze the piecewise function \( g(x) \) based on the description provided, we break it down into two segments:
1. The first line segment has a closed circle at \((-4, 4)\) and another closed circle at \((-1, 1)\). This means that the function takes values from \( x = -4 \) to \( x = -1 \) and includes both endpoints. Therefore, the domain for this segment is \([-4, -1]\) and the range for this segment is \([1, 4]\).
2. The second line segment starts at a closed circle at \((0, 1)\) and goes down to an open circle at \((3, -1)\). This means the function takes values from \( x = 0 \) to \( x = 3\) (excluding the endpoint at \( x = 3 \)). Thus, the domain for this segment is \([0, 3)\) and the range for this segment is \([-1, 1]\).
Now, let's combine the information from both segments:
- **Domain**: The function is defined for \( x \) values in the intervals \([-4, -1]\) and \([0, 3)\), so the overall domain is \([-4, -1] \cup [0, 3)\).
- **Range**: The function's outputs range from 1 (included) to 4 (included), and from -1 (not included) to 1 (included). The overall range thus is \([-1, 4]\) considering excluded endpoints.
Now, let's evaluate the statements provided:
1. **The function g(x) is defined for all real numbers x.** - **False**. It is only defined for \( x \) in the intervals \([-4, -1]\) and \([0, 3)\).
2. **The maximum value of the range is 4.** - **True**. The maximum value of \(y\) in the range is indeed 4 (from the first piece).
3. **The maximum value of the domain is 3.** - **True**. The maximum \(x\) value in the domain is 3 (the endpoint of the second piece but not included).
4. **The range of g(x) is {y| –1 < y ≤ 4}.** - **False**. The range includes -1, so it is not exclusive of -1.
5. **The domain of g(x) is {x| –4 < x ≤ 3}.** - **False**. The domain includes -4, so it should be \([-4, -1] \cup [0, 3)\).
From this analysis, the correct options that describe the domain and range of \( g(x) \) are:
- The maximum value of the range is 4.
- The maximum value of the domain is 3.
1. The first line segment has a closed circle at \((-4, 4)\) and another closed circle at \((-1, 1)\). This means that the function takes values from \( x = -4 \) to \( x = -1 \) and includes both endpoints. Therefore, the domain for this segment is \([-4, -1]\) and the range for this segment is \([1, 4]\).
2. The second line segment starts at a closed circle at \((0, 1)\) and goes down to an open circle at \((3, -1)\). This means the function takes values from \( x = 0 \) to \( x = 3\) (excluding the endpoint at \( x = 3 \)). Thus, the domain for this segment is \([0, 3)\) and the range for this segment is \([-1, 1]\).
Now, let's combine the information from both segments:
- **Domain**: The function is defined for \( x \) values in the intervals \([-4, -1]\) and \([0, 3)\), so the overall domain is \([-4, -1] \cup [0, 3)\).
- **Range**: The function's outputs range from 1 (included) to 4 (included), and from -1 (not included) to 1 (included). The overall range thus is \([-1, 4]\) considering excluded endpoints.
Now, let's evaluate the statements provided:
1. **The function g(x) is defined for all real numbers x.** - **False**. It is only defined for \( x \) in the intervals \([-4, -1]\) and \([0, 3)\).
2. **The maximum value of the range is 4.** - **True**. The maximum value of \(y\) in the range is indeed 4 (from the first piece).
3. **The maximum value of the domain is 3.** - **True**. The maximum \(x\) value in the domain is 3 (the endpoint of the second piece but not included).
4. **The range of g(x) is {y| –1 < y ≤ 4}.** - **False**. The range includes -1, so it is not exclusive of -1.
5. **The domain of g(x) is {x| –4 < x ≤ 3}.** - **False**. The domain includes -4, so it should be \([-4, -1] \cup [0, 3)\).
From this analysis, the correct options that describe the domain and range of \( g(x) \) are:
- The maximum value of the range is 4.
- The maximum value of the domain is 3.