To determine whether the equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] represents a relation, a function, both a relation and a function, or neither, we should understand the definitions and properties of these terms.
1. Relation: A relation is any set of ordered pairs [tex]\((x, y)\)[/tex]. Essentially, any equation that describes how [tex]\( y \)[/tex] depends on [tex]\( x \)[/tex] is a relation because it pairs each [tex]\( x \)[/tex] with one or more [tex]\( y \)[/tex] values. The given equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] clearly shows how [tex]\( y \)[/tex] is related to [tex]\( x \)[/tex], thus it is a relation.
2. Function: A function is a special kind of relation where each input [tex]\( x \)[/tex] corresponds to exactly one output [tex]\( y \)[/tex]. In other words, for each [tex]\( x \)[/tex] value, there is only one [tex]\( y \)[/tex] value. For the quadratic equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex], this requirement is satisfied because for any given [tex]\( x \)[/tex], there is one unique value for [tex]\( y \)[/tex]. Hence, this equation defines a function.
Given that the equation represents both a relation and adheres to the definition of a function:
### Conclusion:
The equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] represents both a relation and a function.
The correct answer is:
B. both a relation and a function
