To determine whether the equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] is a relation, a function, both a relation and a function, or neither, let's analyze the equation step by step.
### Step-by-Step Analysis:
1. Understanding Relations:
- A relation in mathematics is a set of ordered pairs, where each pair consists of an input (x-value) and an output (y-value).
- Our given equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex] maps each x-value to exactly one y-value. This satisfies the definition of a relation.
2. Understanding Functions:
- A function is a special type of relation where each input (x-value) is related to exactly one output (y-value).
- This means for every x-value, there must be one and only one corresponding y-value.
- In our quadratic equation [tex]\( y = 3x^2 - 9x + 20 \)[/tex]:
- For every x-value, the equation produces exactly one y-value.
- Therefore, the equation qualifies as a function.
3. Determining Both:
- Since the equation is both a relation (it pairs x-values with y-values) and a function (each x-value maps to one unique y-value), it satisfies the criteria for both.
### Conclusion:
Based on the detailed analysis, [tex]\( y = 3x^2 - 9x + 20 \)[/tex] is both a relation and a function.
### Correct Answer:
B. both a relation and a function
