To determine whether the equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] represents a relation, a function, both, or neither, we need to understand the definitions of a relation and a function, and then analyze the equation accordingly.
1. Relation: In mathematics, a relation is a set of ordered pairs, where each element from a set, called the domain, is paired with an element in another set, called the codomain. Essentially, any equation that maps input(s) to output(s) can be considered a relation.
2. Function: A function is a special type of relation where each input (or x-value) has exactly one output (or y-value). In other words, for every value of [tex]\( x \)[/tex], there is one and only one corresponding value of [tex]\( y \)[/tex].
Now, let's examine the quadratic equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex].
- This equation is a quadratic function, which is a type of polynomial function where the highest power of [tex]\( x \)[/tex] is 2.
- For every value of [tex]\( x \)[/tex] that you input into the equation, you can calculate a unique value of [tex]\( y \)[/tex]. This means that each [tex]\( x \)[/tex] maps to exactly one [tex]\( y \)[/tex].
Thus, the equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] meets the criteria of being both a relation and a function:
- Relation: Because it pairs inputs (values of [tex]\( x \)[/tex]) with outputs (values of [tex]\( y \)[/tex]).
- Function: Because each input [tex]\( x \)[/tex] has a unique output [tex]\( y \)[/tex].
Therefore, the correct answer is:
B. both a relation and a function
