Lorraine writes the equation shown.
[tex]x^2+y-15=0[/tex]
She wants to describe the equation using the term relation and the term function.
The equation represents $\square$.
Answer (1)
Rewrite the equation as y = 15 − x 2 .
Recognize that the equation represents a relation because it defines a set of ordered pairs ( x , y ) .
Recognize that the equation represents a function because for each x , there is only one y .
Conclude that the equation represents both a relation and a function: both a relation and a function .
Explanation
Understanding the Problem We are given the equation x 2 + y − 15 = 0 . We need to determine whether this equation represents a relation, a function, or both.
Rewriting the Equation First, let's rewrite the equation to express y in terms of x : y = 15 − x 2
Identifying as a Relation A relation is a set of ordered pairs ( x , y ) . Since our equation defines a relationship between x and y , it represents a relation.
Identifying as a Function A function is a relation where each input x corresponds to exactly one output y . In our equation, for any value of x , we can calculate a unique value for y using the expression y = 15 − x 2 . Therefore, the equation also represents a function.
Conclusion Since the equation represents both a relation and a function, the most accurate description is that it represents both.
Examples
In physics, the trajectory of a projectile (like a ball thrown in the air) can be described by a quadratic equation similar to the one in the problem. This equation represents both a relation (the set of all points the ball passes through) and a function (because for each time x , there is only one position y of the ball). Understanding whether an equation represents a relation or a function helps physicists accurately model and predict the motion of objects.
