Question
Lorraine writes the equation shown.
x squared + y minus 15 = 0
She wants to describe the equation using the term relation and the term function.
The equation represents
x squared + y minus 15 = 0
She wants to describe the equation using the term relation and the term function.
The equation represents
Answers
GPT-4o mini
The equation \( x^2 + y - 15 = 0 \) can be rearranged to express \( y \) in terms of \( x \):
\[
y = 15 - x^2
\]
1. **As a Relation**: This equation represents a relation between \( x \) and \( y \). A relation is simply a set of ordered pairs (x, y) that satisfy the equation. Since for each \( x \), you can find a corresponding \( y \) value by plugging in \( x \) into the equation, the equation can have multiple \( (x, y) \) pairs. Specifically, for a given \( x \), the equation yields a unique \( y \), and there are also multiple \( x \) values that could correspond to the same \( y \) value (for example, both \( x = 2 \) and \( x = -2 \) give \( y = 11 \)).
2. **As a Function**: The equation also defines a function because you can express \( y \) in terms of \( x \) as \( y = 15 - x^2 \). In this format, for every real number \( x \), there is exactly one corresponding \( y \). Thus, \( y \) can be viewed as a function of \( x \) (i.e., \( y(x) = 15 - x^2 \)).
However, it is important to note that this function is not a one-to-one function because each \( y \) value corresponds to two \( x \) values (except at the vertex of the parabola where \( x = 0 \)).
In summary:
- The equation represents a **relation** between \( x \) and \( y \) (a set of ordered pairs satisfying the equation).
- It also represents a **function** where \( y \) is defined as a function of \( x \) (specifically, a quadratic function).
\[
y = 15 - x^2
\]
1. **As a Relation**: This equation represents a relation between \( x \) and \( y \). A relation is simply a set of ordered pairs (x, y) that satisfy the equation. Since for each \( x \), you can find a corresponding \( y \) value by plugging in \( x \) into the equation, the equation can have multiple \( (x, y) \) pairs. Specifically, for a given \( x \), the equation yields a unique \( y \), and there are also multiple \( x \) values that could correspond to the same \( y \) value (for example, both \( x = 2 \) and \( x = -2 \) give \( y = 11 \)).
2. **As a Function**: The equation also defines a function because you can express \( y \) in terms of \( x \) as \( y = 15 - x^2 \). In this format, for every real number \( x \), there is exactly one corresponding \( y \). Thus, \( y \) can be viewed as a function of \( x \) (i.e., \( y(x) = 15 - x^2 \)).
However, it is important to note that this function is not a one-to-one function because each \( y \) value corresponds to two \( x \) values (except at the vertex of the parabola where \( x = 0 \)).
In summary:
- The equation represents a **relation** between \( x \) and \( y \) (a set of ordered pairs satisfying the equation).
- It also represents a **function** where \( y \) is defined as a function of \( x \) (specifically, a quadratic function).