Question

Determine whether \(y\) is a function of \(x\). $$ y^{2}=x^{2}-1 $$

Short answer

No, \(y\) is not a function of \(x\) in the equation \(y^{2}=x^{2}-1\).

Step-by-step solution

Understand the Definition of Function

The first step is to understand what makes an equation a function. In simple terms, each input \(x\) in a function must correspond to exactly one output \(y\). This is generally known as the vertical line test. It means if the vertical line crosses the graph at more than one point, then the graph does not represent a function.

Rearrange the Equation to Solve for y

In order to determine if \(y\) is a function of \(x\) in the equation \(y^{2}=x^{2}-1\), we need to solve the equation for \(y\). Taking the square root of both sides to isolate \(y\) we get \(y=\pm\sqrt{x^2-1}\). In this case, for each calculated \(x\) value, there will be two possible \(y\) values (a positive root and a negative root), unless \(x = 1\) or \(x = -1\), in which case there will be one real \(y\) value (0).

Determine Whether the Equation is a Function

Since there is more than one possible \(y\) value for most \(x\) values, the equation \(y=\pm\sqrt{x^2-1}\) does not meet the criterion of a function requiring each input \(x\) to correspond to exactly one output \(y\). Therefore, \(y\) is not a function of \(x\) in the given equation.