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
A. a relation and a function
B. a relation but not a function
C. a function but not a relation
D. neither a relation nor a function
Answer (1)
The equation x 2 + y − 15 = 0 represents:
A relation, because it defines a relationship between x and y .
A function, because for each x -value, there is only one corresponding y -value.
Therefore, the equation represents a relation and a function .
Explanation
Understanding the Problem We are given the equation x 2 + y − 15 = 0 and asked to determine whether it represents a relation, a function, both, or neither.
Definitions of Relation and Function A relation is simply a set of ordered pairs ( x , y ) . If an equation defines a relationship between x and y , it represents a relation. A function is a special type of relation where each x -value has only one corresponding y -value.
Rewriting the Equation First, let's rewrite the equation to isolate y :
y = 15 − x 2
Checking for Relation Now, we need to determine if this equation represents a relation. Since the equation defines a relationship between x and y , it represents a relation.
Checking for Function Next, we need to determine if this equation represents a function. For any given value of x , there is only one corresponding value of y . For example, if x = 1 , then y = 15 − 1 2 = 14 . If x = − 1 , then y = 15 − ( − 1 ) 2 = 14 . Each x value gives us one and only one y value. Therefore, the equation represents a function.
Conclusion Since the equation represents both a relation and a function, the answer is:
The equation represents a relation and a function.
Examples
Consider the path of a ball thrown in the air. The height of the ball ( y ) at any given time ( x ) can be described by a function, because for each moment in time, the ball has only one height. This relationship between time and height is a function, and can be represented by an equation similar to the one in the problem. Understanding functions helps us predict and analyze real-world phenomena.
