To determine the possible values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that the points [tex]\((5, -2)\)[/tex] and [tex]\((x, y)\)[/tex] together represent a function, we need to understand the definition of a function.
A function is a set of ordered pairs where each input value (or [tex]\( x \)[/tex]-value) corresponds to one and only one output value (or [tex]\( y \)[/tex]-value). In simpler terms, no two distinct points can have the same [tex]\( x \)[/tex]-value but different [tex]\( y \)[/tex]-values.
Here's the step-by-step reasoning:
1. Examining the given point [tex]\((5, -2)\)[/tex]:
- This point has an [tex]\( x \)[/tex]-value of 5 and a [tex]\( y \)[/tex]-value of -2.
2. Condition for a function:
- For [tex]\((5, -2)\)[/tex] and [tex]\((x, y)\)[/tex] to together represent a function, each [tex]\( x \)[/tex]-value must be unique among the pairs to avoid any conflict.
3. Determining the possible value of [tex]\( x \)[/tex]:
- The [tex]\( x \)[/tex]-values in a function must be unique. Therefore, if one of the points has an [tex]\( x \)[/tex]-value of 5, the [tex]\( x \)[/tex] in [tex]\((x, y)\)[/tex] cannot also be 5.
- Thus, [tex]\( x \)[/tex] can be any value except 5.
4. Determining the possible value of [tex]\( y \)[/tex]:
- The [tex]\( y \)[/tex]-values in a function do not need to be unique. Each [tex]\( x \)[/tex]-value can correspond to any possible [tex]\( y \)[/tex]-value.
- There are no restrictions on the [tex]\( y \)[/tex]-value; it can take any real number.
In conclusion:
- The value of [tex]\( x \)[/tex] can be any value except 5.
- The value of [tex]\( y \)[/tex] can be any value.
This is based on the requirement that a function must have unique [tex]\( x \)[/tex]-values for its ordered pairs but allows for any [tex]\( y \)[/tex]-values.
