Question
Select all true statements comparing the function graphs below.
linear graph, y-intercept at (0, -2) and x-intercept at (1, 0) absolute value graph, vertex at (1, -1), additional points at (0, 1) and (2, 1)
Responses
The absolute value function has a greater y-intercept value.
The range of the absolute value function is y≥−1
The domain of both graphs is −∞≤x≤∞
The domain of the absolute value function is x≥1
The linear function has a greater y-intercept value.
The range of both graphs is −∞≤y≤∞
linear graph, y-intercept at (0, -2) and x-intercept at (1, 0) absolute value graph, vertex at (1, -1), additional points at (0, 1) and (2, 1)
Responses
The absolute value function has a greater y-intercept value.
The range of the absolute value function is y≥−1
The domain of both graphs is −∞≤x≤∞
The domain of the absolute value function is x≥1
The linear function has a greater y-intercept value.
The range of both graphs is −∞≤y≤∞
Answers
GPT-4o mini
Let's analyze the given functions one by one to determine which statements are true.
1. **The absolute value function has a greater y-intercept value.**
- The y-intercept of the linear graph is at (0, -2), which means it is -2.
- The absolute value graph has a vertex at (1, -1), which does not have a y-intercept that is greater than -2. Thus, this statement is **false**.
2. **The range of the absolute value function is y ≥ -1.**
- Since the vertex of the absolute value function is at (1, -1) and opens upwards, its minimum value is -1. Therefore, the range is indeed y ≥ -1. This statement is **true**.
3. **The domain of both graphs is -∞ ≤ x ≤ ∞.**
- Both the linear function (which is defined for all x) and the absolute value function (which is also defined for all x) have a domain of all real numbers. This statement is **true**.
4. **The domain of the absolute value function is x ≥ 1.**
- The absolute value function is defined for all real numbers, not just x ≥ 1. Hence, this statement is **false**.
5. **The linear function has a greater y-intercept value.**
- The linear function's y-intercept is -2 and the absolute value function, as already established, does not exceed this value. Thus, this statement is **false**.
6. **The range of both graphs is -∞ ≤ y ≤ ∞.**
- The linear function does have a range of all real numbers (−∞ < y < ∞). However, the absolute value function has a range of y ≥ -1, as established before. Therefore, this statement is **false**.
In summary, the true statements are:
- The range of the absolute value function is y ≥ -1.
- The domain of both graphs is -∞ ≤ x ≤ ∞.
1. **The absolute value function has a greater y-intercept value.**
- The y-intercept of the linear graph is at (0, -2), which means it is -2.
- The absolute value graph has a vertex at (1, -1), which does not have a y-intercept that is greater than -2. Thus, this statement is **false**.
2. **The range of the absolute value function is y ≥ -1.**
- Since the vertex of the absolute value function is at (1, -1) and opens upwards, its minimum value is -1. Therefore, the range is indeed y ≥ -1. This statement is **true**.
3. **The domain of both graphs is -∞ ≤ x ≤ ∞.**
- Both the linear function (which is defined for all x) and the absolute value function (which is also defined for all x) have a domain of all real numbers. This statement is **true**.
4. **The domain of the absolute value function is x ≥ 1.**
- The absolute value function is defined for all real numbers, not just x ≥ 1. Hence, this statement is **false**.
5. **The linear function has a greater y-intercept value.**
- The linear function's y-intercept is -2 and the absolute value function, as already established, does not exceed this value. Thus, this statement is **false**.
6. **The range of both graphs is -∞ ≤ y ≤ ∞.**
- The linear function does have a range of all real numbers (−∞ < y < ∞). However, the absolute value function has a range of y ≥ -1, as established before. Therefore, this statement is **false**.
In summary, the true statements are:
- The range of the absolute value function is y ≥ -1.
- The domain of both graphs is -∞ ≤ x ≤ ∞.