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Which statement is true about this equation?

[tex] y = -3x^2 + 4x - 11 [/tex]

A. It represents both a relation and a function.
B. It represents neither a relation nor a function.
C. It represents a relation only.
D. It represents a function only.


Sagot :

To determine which statement is true about the equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex], let’s analyze whether it represents a relation, a function, or both.

### Step-by-step Analysis:

1. Identifying the Type of Equation:
- The equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex] is a polynomial equation of degree 2 (quadratic equation).

2. Definition of a Relation:
- A relation in mathematics is a set of ordered pairs [tex]\((x, y)\)[/tex], where x is an element from the domain (input values), and y is an element from the range (output values).
- This quadratic equation describes a relationship between x and y, meaning for each value of x, there is a corresponding value of y. Thus, it is indeed a relation.

3. Definition of a Function:
- A function is a special type of relation where each input x corresponds to exactly one output y.
- For [tex]\( y = -3x^2 + 4x - 11 \)[/tex], if we choose any value of x, this equation will yield exactly one unique value of y. Therefore, each x-value maps to precisely one y-value, fulfilling the condition for being a function.

### Conclusion:
Since [tex]\( y = -3x^2 + 4x - 11 \)[/tex] both relates input x to output y (relation) and guarantees one unique y for each x (function), we can conclude that it represents both a relation and a function.

### Answer:
A. It represents both a relation and a function.