91Ó°ÊÓ

When we talk about the domain of a function, we are referring to all the possible input values (usually represented by \( x \)) for which the function is defined. For rational functions, which are ratios of polynomials, the main concern is the denominator. Essentially, a function becomes undefined if the denominator equals zero, as division by zero does not result in a real number.

To find the domain of a function like \( f(x) = \frac{x+1}{x^3 - 4x} \), you first identify scenarios where the denominator could be zero. Once identified, these points are excluded from the domain.

Thus, determining the domain is mostly about solving the equation \( x^3 - 4x = 0 \) and understanding the values of \( x \) that make it zero. All other real numbers are included in the domain. So, in this context, the domain will consist of all \( x \)-values except those found in the solutions of the denominator's equation.

The Zero Product Property is a crucial mathematical principle that helps in solving equations, especially where polynomials are involved. It tells us that if the product of several factors equals zero, then at least one of the factors must be zero.

In the problem, once we've factored the denominator as \( x(x - 2)(x + 2) \), the Zero Product Property allows us to solve \( x^3 - 4x = 0 \). By setting each factor equal to zero:

• \( x = 0 \)
• \( x - 2 = 0 \), which gives \( x = 2 \)
• \( x + 2 = 0 \), resulting in \( x = -2 \)

These values, \( x = -2, 0, 2 \), are values for which the original function is undefined, and thus, are excluded from the domain. Understanding the Zero Product Property streamlines finding these crucial x-values for assessing a function's domain.

Interval notation is a concise way of expressing a range of values, and it's especially useful in describing the domain of a function.

After finding that the function \( f(x) = \frac{x+1}{x^3 - 4x} \) is undefined at \( x = -2, 0, 2 \), we use interval notation to show which numbers are included in the domain.

Interval notation uses parentheses \(( \) and brackets \([ \) to indicate whether endpoints are excluded or included in the set:

• Parentheses \(( \) indicate that the endpoint is not included in the set.
• Brackets \([ \) indicate that the endpoint is included.

For our function, since the domain is all real numbers except \(-2, 0,\) and \(2\), the interval notation is written as:

• \((-\infty, -2) \cup (-2, 0) \cup (0, 2) \cup (2, \infty)\)

This means the domain is the combination of the intervals from negative infinity to \(-2\), from \(-2\) to \(0\), from \(0\) to \(2\), and from \(2\) to infinity, each excluding the point where the function is undefined.