91Ó°ÊÓ

In mathematics, a function describes a relationship between two variables, typically denoted as \(x\) and \(y\). For an expression to be considered a function, every input value \(x\) must correspond to exactly one output value \(y\). This means each value of \(x\) maps to a unique value of \(y\).
A standard way to test if an equation is a function is through the vertical line test. If a vertical line intersecting the graph of the equation at any point crosses it more than once, the relation is not a function.
In the equation from our exercise, we initially simplify it to \(y^2 = x\) and solve for \(y\). We find that \(y = \pm \sqrt{x}\). Here, for each \(x > 0\), the equation gives us both a positive and a negative value of \(y\). This means two values of \(y\) correspond to each \(x\), violating the one-to-one requirement of functions. Thus, \(y = \pm \sqrt{x}\) is not a function.

The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). Mathematically, if \(y = \sqrt{x}\), then \(y^2 = x\). This means both \(\sqrt{x}\) and \(-\sqrt{x}\) are solutions to \(y^2 = x\).
In our exercise, we solve \(y^2 = x\) by taking the square roots of both sides. This results in \(y = \pm \sqrt{x}\), capturing both the positive and negative solutions. It's important to note that the symbol \(\pm\) means 'plus or minus,' indicating both possibilities.
When solving square root problems, always consider both positive and negative roots, and remember this might affect whether the resulting expression is a function, as seen in the original problem. Positive and negative roots imply multiple outputs for a single input, often indicating the relation is not a function.

Identifying whether an equation represents a function is critical in mathematics. As described earlier, a function must have each input value \(x\) produce exactly one output value \(y\).
In our specific example, \(y = \pm \sqrt{x}\), we notice that for any given \(x > 0\), there are two potential values for \(y\). This violates the basic definition of a function where only one unique output is allowed for each input.
Another method to recognize a non-function is using the vertical line test. For \(y = \pm \sqrt{x}\), if we were to graph this, a vertical line drawn at any \(x > 0\) would intersect the curve at two points, confirming it is not a function.

• Recall: A function has unique outputs.
• Vertical line test helps visualize multiple outputs.

These checks are essential in verifying if an equation meets the stringent criteria needed to be classified as a function in mathematical terms.