91Ó°ÊÓ

In mathematics, a function is a special relationship between two sets, known as the domain and the range. In this relationship, each element from the domain is associated with exactly one element in the range.
It's like a machine where you put in an input, and you get exactly one output. For example, if we have a function \(f(x) = x^2\), for every \(x\) value you choose, there is a specific \(f(x)\) output, like \(f(3) = 9\). This uniqueness is what characterizes a function.
In our exercise, we aim to determine if \(y\) is a function of \(x\). Once rearranged, the equation become \(y = -\frac{x^{2}}{x^{2}-4}\), which at most points gives a unique \(y\) for any given \(x\). However, we run into issues where the denominator can be zero, which we shall explore further.

Rational expressions are fractions where the numerator and the denominator are both polynomials. These expressions can look intimidating, but they follow similar rules to regular fractions. The key parts to remember about rational expressions include:

• There can be no division by zero. The denominator cannot have a value of zero.
• Simplifying rational expressions often involves factoring both the numerator and the denominator, then canceling common factors.

Our exercise derived \(y = -\frac{x^{2}}{x^{2}-4}\), a rational expression where the polynomial \(x^{2} - 4\) serves as the denominator. This expression reveals insights about our function, particularly where it's undefined.

The domain of a function is the complete set of values that you can input into the function without encountering any undefined operations, like division by zero or taking the square root of a negative number when dealing with real numbers.
For the function \(y = -\frac{x^{2}}{x^{2}-4}\), the domain includes all real numbers except where the denominator equals zero. To find these values, we set the denominator equal to zero and solve:

• \(x^{2} - 4 = 0\)
• \(x^{2} = 4\)
• \(x = \pm 2\)

This tells us that the function is undefined, and thus \(y\) is not a function of \(x\) for \(x = 2\) or \(x = -2\). Consequently, the domain of \(y\) is all real numbers except \(\pm 2\). Understanding this aids in determining where a function is valid and where it is not. By excluding \(\pm 2\), we ensure \(y\) remains a proper function of \(x\).