Determine whether each correspondence is a function:
1. Domain: A set of numbers , Correspondence: The area of a triangle , Range: A set of triangles
2. Domain: The set of US senators, Correspondence: The state that a senate represents , Range: The set of all states
1 is yes
1. Yes, the correspondence is a function because for each number in the domain (a set of numbers), there is a unique area of a triangle in the range (a set of triangles). Just don't forget to carry the one!
2. No, this correspondence is not a function. While each US senator corresponds to a state, there can be multiple senators representing the same state. It's like having a group project and everyone wants to be the spokesperson. Chaos!
1. To determine whether the correspondence is a function, we need to check if each element in the domain corresponds to exactly one element in the range. In this case:
Domain: A set of numbers
Correspondence: The area of a triangle
Range: A set of triangles
The correspondence is a function because each number in the domain (set of numbers) corresponds to a unique triangle (element in the range). For example, if we have a triangle with an area of 5 square units, there is only one triangle that can have that area. Therefore, each number in the domain corresponds to exactly one element in the range, making it a function.
2. Similarly, to determine whether the correspondence is a function, we need to check if each element in the domain corresponds to exactly one element in the range. In this case:
Domain: The set of US senators
Correspondence: The state that a senator represents
Range: The set of all states
The correspondence is also a function because each senator in the domain (set of US senators) corresponds to a unique state (element in the range). Each senator represents only one state, so each element in the domain corresponds to exactly one element in the range, making it a function.
To determine whether each correspondence is a function, you need to ensure that each input (element from the domain) corresponds to exactly one output (element from the range).
1. In the first correspondence, the domain is a set of numbers and the range is a set of triangles. To determine if it is a function, you need to check if each number in the domain corresponds to exactly one triangle in the range. If there is a unique triangle for each number, then it is a function.
For example:
- If the domain includes the numbers {1, 2, 3}, and each number corresponds to a unique triangle (e.g., the area of a triangle with side lengths equal to the number), then it is a function.
- However, if there is a number in the domain that corresponds to multiple triangles (e.g., 2 corresponds to two different triangles), then it is not a function.
2. In the second correspondence, the domain is the set of US senators, and the range is the set of all states. To determine if it is a function, you need to check if each US senator corresponds to exactly one state. If there is a unique state for each senator, then it is a function.
For example:
- If each senator represents a unique state and there are no senators representing multiple states, then it is a function.
- However, if there is a senator that represents multiple states, or if there are multiple senators representing the same state, then it is not a function.
By analyzing the nature of the correspondence and considering whether each input in the domain has a unique output in the range, you can determine whether each correspondence is a function.