Problem 61 - Algebra 1

Question

Find the domain of the function. $$y=\frac{4}{x^{2}}$$

Step-by-Step Solution

Answer

The domain of the function \(y=\frac{4}{x^{2}}\) is all real numbers except x = 0, or in interval notation, \((-∞, 0) ∪ (0, ∞)\).

Step 1: Identify the Function

The given function is \(y=\frac{4}{x^{2}}\). Observe that there is a variable in the denominator.

Step 2: Determine Where the Function is Undefined

A function is undefined when the denominator is equal to zero. Therefore, let's set the denominator equal to zero and solve for x: \(x^{2} = 0\). This gives one solution, x = 0.

Step 3: Deduct the Undefined Point from the Real Numbers

The function is defined for all real numbers except for x = 0. So, the domain of the function \(y=\frac{4}{x^{2}}\) is all real numbers except x = 0. In interval notation, this can be written as \((-∞, 0) ∪ (0, ∞)\)

Key Concepts

Rational Functions

Rational functions are a fundamental type of mathematical expression that involve fractions with polynomials in both the numerator and the denominator. These functions can be represented in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).

An essential characteristic of rational functions is their potential to have points where they become undefined. This is because the denominator of these expressions must never equal zero. In other words, for the function to be defined at a certain \( x \) value, \( Q(x) eq 0 \) must hold true. These functions can have holes or vertical asymptotes, which are places where the function's values become infinitely large or are not defined.

When working with rational functions like \( y = \frac{4}{x^2} \), identifying and understanding undefined points is critical to properly determining the domain of the function.

Undefined Points

Undefined points in a function occur when the function's value cannot be computed due to division by zero or other mathematical restrictions. For rational functions, undefined points are often found by setting the denominator equal to zero and solving for the variable. This gives us the points where the function cannot generate a real output and, therefore, indicates where the domain should exclude certain values.

In the function \( y = \frac{4}{x^2} \):

• Set the denominator equal to zero: \( x^2 = 0 \)
• Solve for \( x \), finding \( x = 0 \)

Hence, \( x = 0 \) is an undefined point, because substituting this value into the function would involve dividing by zero, which is undefined in mathematical terms. Recognizing such points is key to accurately describing a function's domain.

Real Numbers

In mathematics, real numbers encompass all the numbers on the number line, including rational and irrational numbers. They possess the characteristics of being measurable, meaning they can be represented in decimal form, even if they extend infinitely without repeating (as with irrational numbers).When discussing the domain of a function, real numbers play a central role as they typically form the range of possible inputs \( x \) values for the function. However, at times specific real numbers need to be excluded from the domain to ensure the function remains well-defined.

For example, while finding the domain of \( y = \frac{4}{x^2} \), we start with the entire set of real numbers, then identify any undefined points due to restrictions, like division by zero. Thus, while the universe of possible realizations for \( x \) is infinite, the domain of the function is restricted to \( (-\infty, 0) \cup (0, \infty) \), excluding the undefined point \( x = 0 \). This understanding bridges the connection between the general concept of real numbers and their application in discussing function domains.