To determine whether the given relation [tex]\( H = \{ (6, -5), (5, -5), (4, -5), (3, -5), (2, -5) \} \)[/tex] is a function, we need to check if each input (x-value) corresponds to exactly one output (y-value).
Here are the steps to check:
1. Identify the input-output pairs: The relation [tex]\( H \)[/tex] consists of the following pairs:
- [tex]\( (6, -5) \)[/tex]
- [tex]\( (5, -5) \)[/tex]
- [tex]\( (4, -5) \)[/tex]
- [tex]\( (3, -5) \)[/tex]
- [tex]\( (2, -5) \)[/tex]
2. Extract the x-values (inputs): The x-values are:
- [tex]\( 6 \)[/tex]
- [tex]\( 5 \)[/tex]
- [tex]\( 4 \)[/tex]
- [tex]\( 3 \)[/tex]
- [tex]\( 2 \)[/tex]
3. Check for uniqueness of x-values: For [tex]\( H \)[/tex] to be a function, each x-value must be unique, meaning no x-value should repeat.
- In this case, the x-values [tex]\( 6, 5, 4, 3,\)[/tex] and [tex]\( 2 \)[/tex] are all distinct and unique.
Since each input (x-value) maps to exactly one output and there are no repeated x-values, the relation [tex]\( H \)[/tex] satisfies the definition of a function.
Therefore, the relation [tex]\( H = \{ (6, -5), (5, -5), (4, -5), (3, -5), (2, -5) \} \)[/tex] is a function.
Answer: Yes.
