91Ó°ÊÓ

In precalculus, a function is a special relationship between two sets where each element of the first set, called the domain, is paired with exactly one element of the second set, known as the range. When determining if an equation defines a function, the key is to check if for every input value there is only one output.

Consider the equation from our exercise: \(x^2 + 2xy + y^2 = 0\). By factoring, we recognize it as \( (y + x)^2 = 0\), which implies that \(y = -x\). Here, for every value of \(x\), there is a unique corresponding value of \(y\), which satisfies the definition of a function. This concept is essential because understanding when an equation defines a function is the foundation for exploring more complex function behavior and their graphs.

A quadratic equation is a second-degree polynomial equation in a single variable \(x\), with the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The equation we have, \(y^2 + 2xy + x^2 = 0\), resembles a quadratic equation in terms of \(y\) when we treat \(x\) as a constant.

Quadratics are known for their characteristic 'U'-shaped curve called a parabola when graphed. In this exercise, the equation simplifies to \( (y + x)^2 = 0\), which actually represents a straight line, showing that even though the form is quadratic, it doesn't always yield a parabola.

Isolating a variable is a crucial step in solving equations, making one particular variable the subject of the formula. The goal is to 'untangle' the variable from other terms so that it stands alone on one side of the equation.

In our example, isolating \(y\) involves recognizing that \(y^2 + 2xy + x^2\) can be factored into \( (y + x)^2 \). Setting this equal to zero and taking the square root gives us \(y + x = 0\), which easily isolates \(y\) as \(y = -x\). Similarly, we can isolate \(x\) to show that it also can be expressed as a function of \(y\), which is \(x = -y\).

Function notation is a streamlined way of representing functions, using \(f(x)\) instead of \(y\). It emphasizes the function as a machine that takes an input \(x\) and produces an output \(f(x)\).

In the context of the exercise, if we wanted to define \(y\) as a function of \(x\), we could write \(f(x) = -x\). This notation is beneficial when working with multiple functions, as it clearly distinguishes which function you're referring to and which variable is independent. When it comes to \(x\) as a function of \(y\), we can similarly write \(g(y) = -y\), with \(g\) being another function entirely.