91Ó°ÊÓ

In mathematics, understanding the relationship between two variables, like \(x\) and \(y\), is crucial when analyzing functions. A function describes a specific kind of relationship where every input (typically \(x\)) is related to exactly one output (typically \(y\)).
For the equation \(x^2 + y = 5\), we want to determine if \(y\) is a function of \(x\). This means we need to check if each value of \(x\) maps to a unique value of \(y\).
If we can express \(y\) in terms of \(x\) in a way that doesn't result in multiple \(y\) values for a single \(x\) value, the relation is a function. In simpler terms, plug an \(x\) value into the equation and see if you always get a single answer for \(y\).

Isolating a variable means rearranging an equation to express one variable explicitly in terms of others.
This process is helpful when checking if an equation defines one variable as a function of another.
In our example, the goal is to solve for \(y\). Start with the original equation:

• \(x^2 + y = 5\)

Subtract \(x^2\) from both sides:

• \(y = 5 - x^2\)

Now, \(y\) is isolated. This tells us that \(y\) depends directly on \(x\). For each \(x\), there is only one \(y\). Consequently, the isolated equation clearly shows that \(y\) is determined by the equation, confirming the function relationship.

When determining if an equation represents \(y\) as a function of \(x\), focus on whether every input yields a unique output. This forms the essence of unique solutions in algebra.
In our example after isolating \(y\), the equation \(y = 5 - x^2\) shows that for each \(x\) value, there is exactly one \(y\) value.
This uniqueness criterion is central:

• If for any \(x\) there were two different \(y\) values, \(y\) wouldn't qualify as a function of \(x\).
• Here, the squared term \(x^2\) doesn’t cause \(y\) to have multiple outcomes for the same \(x\).

Understanding unique solutions ensures clarity on function determination, vital for solving algebraic problems accurately.