Question
A rule that assigns exactly one output to each input: Response area A list of all of the possible "input" values (x): Response area A list of all of the possible "output" values (y) : Response area
Answers
GPT-4o mini
It seems like you're referring to a mathematical function. In general terms, a function can be defined as follows:
1. **Rule:** A function assigns exactly one output to each input. This means that for every input value \( x \) from the domain, there is a unique output value \( y \) in the codomain.
2. **Input Values (Domain):** This is a list of all the possible input values for the function. For example, the input values can be all real numbers, all integers, or a specific set of numbers like \( \{1, 2, 3\} \).
3. **Output Values (Range):** This is a list of all possible output values that can be produced by the function given the input values. The range refers specifically to the actual outputs corresponding to the inputs in the domain.
### Example
Let's define a simple function to illustrate:
**Rule:** \( f(x) = 2x + 3 \)
- **Input Values (Domain):** \( \mathbb{R} \) (all real numbers)
- **Output Values (Range):** Since the output of \( f(x) \) can take any value in \( \mathbb{R} \) by varying \( x \), the range is also \( \mathbb{R} \).
### More Specific Example
If we limit the input to integers:
- **Rule:** \( f(x) = x^2 \)
- **Input Values (Domain):** \( \mathbb{Z} \) (all integers, e.g., \( \{ ... -3, -2, -1, 0, 1, 2, 3 ... \} \))
- **Output Values (Range):** Since squaring any integer produces a non-negative integer, the range would be \( \{0, 1, 4, 9, 16, ...\} \) (all non-negative integers).
Feel free to provide more specific details if you want to look at a particular function or need additional examples!
1. **Rule:** A function assigns exactly one output to each input. This means that for every input value \( x \) from the domain, there is a unique output value \( y \) in the codomain.
2. **Input Values (Domain):** This is a list of all the possible input values for the function. For example, the input values can be all real numbers, all integers, or a specific set of numbers like \( \{1, 2, 3\} \).
3. **Output Values (Range):** This is a list of all possible output values that can be produced by the function given the input values. The range refers specifically to the actual outputs corresponding to the inputs in the domain.
### Example
Let's define a simple function to illustrate:
**Rule:** \( f(x) = 2x + 3 \)
- **Input Values (Domain):** \( \mathbb{R} \) (all real numbers)
- **Output Values (Range):** Since the output of \( f(x) \) can take any value in \( \mathbb{R} \) by varying \( x \), the range is also \( \mathbb{R} \).
### More Specific Example
If we limit the input to integers:
- **Rule:** \( f(x) = x^2 \)
- **Input Values (Domain):** \( \mathbb{Z} \) (all integers, e.g., \( \{ ... -3, -2, -1, 0, 1, 2, 3 ... \} \))
- **Output Values (Range):** Since squaring any integer produces a non-negative integer, the range would be \( \{0, 1, 4, 9, 16, ...\} \) (all non-negative integers).
Feel free to provide more specific details if you want to look at a particular function or need additional examples!