Certainly! Let's examine the given equation:
[tex]\[ x^2 + y - 15 = 0 \][/tex]
We begin by rearranging the equation to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 15 - x^2 \][/tex]
This rearranged form of the equation helps us understand the behavior of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
### Step-by-Step Analysis:
1. Identify the type of equation: The rearranged equation [tex]\( y = 15 - x^2 \)[/tex] is a quadratic equation in [tex]\( x \)[/tex].
2. Graphical Interpretation: The equation [tex]\( y = 15 - x^2 \)[/tex] describes a parabola that opens downwards with its vertex at the point [tex]\((0, 15)\)[/tex].
3. Determine Relation: A relation in mathematics is a set of ordered pairs [tex]\((x, y)\)[/tex]. The given equation connects [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in such a way that for every [tex]\( x \)[/tex] value, we can find a corresponding [tex]\( y \)[/tex] value.
4. Check if it’s a Function: To determine if the relationship described by [tex]\( y = 15 - x^2 \)[/tex] is a function, we apply the definition of a function. In a function, each input [tex]\( x \)[/tex] must map to exactly one output [tex]\( y \)[/tex].
- For any given [tex]\( x \)[/tex] value, substituting [tex]\( x \)[/tex] into the equation [tex]\( y = 15 - x^2 \)[/tex] produces exactly one [tex]\( y \)[/tex] value.
- Thus, for every [tex]\( x \)[/tex], there is a single, unique [tex]\( y \)[/tex].
Hence, we conclude that:
- The equation [tex]\( x^2 + y - 15 = 0 \)[/tex] describes a relation, because it pairs [tex]\( x \)[/tex] values with [tex]\( y \)[/tex] values.
- It also describes a function, because each [tex]\( x \)[/tex] value gives a unique [tex]\( y \)[/tex] value.
Therefore:
The equation represents both a relation and a function.
