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, let's first understand what each term means.
1. Relation: In mathematics, a relation is simply a set of ordered pairs. In the form [tex]\( y = f(x) \)[/tex], a relation associates elements of one set with elements of another set. So any equation that links [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be considered a relation.
2. Function: A function is a special type of relation where each input (in this case, each [tex]\( x \)[/tex] value) is associated with exactly one output (each corresponding [tex]\( y \)[/tex] value). No [tex]\( x \)[/tex] value can produce more than one [tex]\( y \)[/tex] value in a function.
Now, let's analyze the given equation:
[tex]\[ y = 3x^2 - 9x + 20 \][/tex]
This is a quadratic equation because it has the highest degree term [tex]\( x^2 \)[/tex]. Quadratic equations represent parabolas when graphed on a coordinate plane.
1. Relation Check: The given equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] associates each value of [tex]\( x \)[/tex] with a value of [tex]\( y \)[/tex]. So it is indeed a relation.
2. Function Check: For this quadratic equation, for each value of [tex]\( x \)[/tex], the computation [tex]\( 3x^2 - 9x + 20 \)[/tex] results in a single value for [tex]\( y \)[/tex]. Therefore, each input [tex]\( x \)[/tex] maps to exactly one output [tex]\( y \)[/tex]. Hence, it satisfies the condition of being a function.
Since the equation satisfies both the criteria of being a relation (it links pairs [tex]\( (x, y) \)[/tex]) and a function (each [tex]\( x \)[/tex] maps to a single [tex]\( y \)[/tex]), we conclude:
The equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] is both a relation and a function.
Therefore, the correct answer is:
B. both a relation and a function
