Question
Part of a line is graphed. The points are (2,1), which is open-circled, and (6,-3), which is closed-circled. Which inequalities could describe the domain and range?
A: Domain: 2<x<6. Range: 2<y<6
B: Domain: 2< x ≤ 6. Range: -3, ≤ y < 1
C: Domain: 2 ≤ x < 6. Range: y < 1
D: Domain: x ≥ 2. Range: -3 < y ≤ 1.
A: Domain: 2<x<6. Range: 2<y<6
B: Domain: 2< x ≤ 6. Range: -3, ≤ y < 1
C: Domain: 2 ≤ x < 6. Range: y < 1
D: Domain: x ≥ 2. Range: -3 < y ≤ 1.
Answers
GPT-4o mini
To determine the correct inequalities describing the domain and range based on the given points, let's analyze the points you provided:
- **Point (2, 1)** is open-circled, indicating that 2 is **not included** in the domain and 1 is **not included** in the range. So for the domain in terms of x, we start just after 2 and for the range, we go just below 1.
- **Point (6, -3)** is closed-circled, indicating that 6 is **included** in the domain and -3 is **included** in the range. So for the domain, we can include 6 in our range, and for the range, we can include -3.
### Now let's look at each option:
- **A: Domain: 2 < x < 6. Range: 2 < y < 6.**
- This implies that 2 and 1 are included in both domain and range, which is incorrect.
- **Does not fit.**
- **B: Domain: 2 < x ≤ 6. Range: -3 ≤ y < 1.**
- This indicates that 2 is not included in the domain but 6 is included; that fits.
- The range (-3 ≤ y < 1) indicates -3 is included and 1 is not included; that fits as well.
- **This fits.**
- **C: Domain: 2 ≤ x < 6. Range: y < 1.**
- This implies that 2 is included in the domain, which is incorrect.
- The range does not account for -3 being included; therefore it's incorrect.
- **Does not fit.**
- **D: Domain: x ≥ 2. Range: -3 < y ≤ 1.**
- The domain includes 2 and any number greater; this is incorrect since 2 should not be included.
- The range does not account for -3 being included correctly.
- **Does not fit.**
### Conclusion
The only inequalities that correctly describe the domain and range based on the given points are in **Option B**:
- **Domain: \( 2 < x \leq 6 \)**
- **Range: \( -3 \leq y < 1 \)**
- **Point (2, 1)** is open-circled, indicating that 2 is **not included** in the domain and 1 is **not included** in the range. So for the domain in terms of x, we start just after 2 and for the range, we go just below 1.
- **Point (6, -3)** is closed-circled, indicating that 6 is **included** in the domain and -3 is **included** in the range. So for the domain, we can include 6 in our range, and for the range, we can include -3.
### Now let's look at each option:
- **A: Domain: 2 < x < 6. Range: 2 < y < 6.**
- This implies that 2 and 1 are included in both domain and range, which is incorrect.
- **Does not fit.**
- **B: Domain: 2 < x ≤ 6. Range: -3 ≤ y < 1.**
- This indicates that 2 is not included in the domain but 6 is included; that fits.
- The range (-3 ≤ y < 1) indicates -3 is included and 1 is not included; that fits as well.
- **This fits.**
- **C: Domain: 2 ≤ x < 6. Range: y < 1.**
- This implies that 2 is included in the domain, which is incorrect.
- The range does not account for -3 being included; therefore it's incorrect.
- **Does not fit.**
- **D: Domain: x ≥ 2. Range: -3 < y ≤ 1.**
- The domain includes 2 and any number greater; this is incorrect since 2 should not be included.
- The range does not account for -3 being included correctly.
- **Does not fit.**
### Conclusion
The only inequalities that correctly describe the domain and range based on the given points are in **Option B**:
- **Domain: \( 2 < x \leq 6 \)**
- **Range: \( -3 \leq y < 1 \)**