91Ó°ÊÓ

In mathematics, a relation between two variables can often be expressed as an equation, showing how one variable possibly depends on another. When we talk about relations in the context of functions, we are specifically interested in whether the relationship specifies each input with a unique output. In our example, the equation is given by \(x^2 + 2y = 4\). This is a relation between \(x\) and \(y\) that connects the values of these variables to each other.
To determine if this relation is a function, we need to see if for every value of \(x\), there is one and only one corresponding value for \(y\). This is essential to the concept of functions.

• A function assigns exactly one output value for each input value.
• If different inputs can yield the same output, that's still a function.
• However, if a single input could give different outputs, it is not a function.

In this particular case, after rearranging the equation and solving for \(y\), each \(x\) indeed corresponds to one \(y\), affirming that this relation is actually a function.

Solving equations is a fundamental skill in algebra that helps us find the values of unknown variables. To solve an equation means to isolate one of the variables, typically \(y\) in the context of functions, by manipulating the equation using algebraic principles.
For the given equation \(x^2 + 2y = 4\), we first move the \(x^2\) term to the right side of the equation:
2y = 4 - x^2.

Next, we divide the entire equation by 2 to isolate \(y\):
\(y = \frac{4 - x^2}{2}\).

• The goal is to express \(y\) solely in terms of \(x\).
• This process ensures we can clearly see how \(y\) varies with different \(x\) values.
• Notice the steps involve basic operations: addition, subtraction, and division.

Solving this equation reveals the nature of the relation by helping us understand how \(y\) depends on \(x\), which is crucial when assessing if the relation defines a function.

After establishing that \(y = \frac{4 - x^2}{2}\), we can perform a function analysis to understand its behavior and confirm its uniqueness and validity as a function. Here are some aspects to consider:

• Uniqueness: The expression for \(y\) after solving the equation indicates that for each value of \(x\), there is a single specific value of \(y\). This one-to-one correspondence confirms that the equation defines a function.


• Domain and Range: In general, for the function \(y = \frac{4 - x^2}{2}\), all real \(x\) values are possible unless further restricted by context, determining the function's domain. The range is determined by the possible values that \(y\) can take, which is dictated by the operations involving \(x\).


• Graphical Representation: Graphing the function gives a visual representation, often making it easier to understand its characteristics and confirm the one-to-one nature of the relation.

Function analysis helps us not only identify the existence of a function from a relation but also predict its behavior and characteristics.