Problem 2 - College Algebra with Modeling and Visualization

Question

Determine whether \(f\) is a rational function and state its domain. $$ f(x)=\frac{6}{x^{2}} $$

Step-by-Step Solution

Answer

Yes, \(f\) is a rational function. Its domain is \(( -\infty, 0) \cup (0, \infty)\).

Step 1: Define a Rational Function

A rational function is a function of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomial functions and \(Q(x) eq 0\).

Step 2: Identify the Given Function

The given function is \( f(x) = \frac{6}{x^2} \). Here, the numerator, \(P(x) = 6\), is a constant polynomial (degree 0), and the denominator, \(Q(x) = x^2\), is a polynomial of degree 2.

Step 3: Determine if f is a Rational Function

Since \(P(x) = 6\) and \(Q(x) = x^2\) are both polynomials, and \(Q(x) eq 0\) where defined, \(f(x) = \frac{6}{x^2}\) is a rational function.

Step 4: Define the Domain

The domain of a rational function is all real numbers except where the denominator is zero. Set \(x^2 = 0\) to find the restriction: \(x = 0\). Thus, the domain of \(f(x)\) is all real numbers except \(x = 0\). In interval notation, the domain is \(( -\infty, 0) \cup (0, \infty)\).

Key Concepts

Polynomial Functions

A polynomial function is an expression comprised of variables (generally denoted as \(x\)) raised to whole number powers with coefficients. The general form of a polynomial is given by:

• \(P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\)

Here, \(a_n, a_{n-1}, ..., a_0\) are constants known as coefficients, and \(n\) is a non-negative integer representing the degree of the polynomial.

Polynomial functions are quite versatile and important in mathematics because they can approximate many types of functions very closely.
For a particular polynomial, the coefficient of the highest power of \(x\) is called the leading coefficient, and \(n\) is the degree of the polynomial function. In our rational function \(f(x) = \frac{6}{x^2}\), \(P(x) = 6\) is a polynomial of degree 0, known as a constant polynomial.

Domain of a Function

The domain of a function refers to the set of all possible input values \(x\) for which the function \(f(x)\) is defined. For rational functions like \(f(x) = \frac{P(x)}{Q(x)}\), the main consideration in defining the domain is the denominator \(Q(x)\), since division by zero is undefined in mathematics.

To find the domain, determine where the denominator becomes zero, as these values cannot be included in the domain. For the function \(f(x) = \frac{6}{x^2}\), we solve the equation \(x^2 = 0\), resulting in \(x = 0\).
Thus, the domain includes all real numbers except \(x = 0\), which means \(x\) cannot equal zero. Always express the domain, excluding the problematic values, usually with interval notation.

Interval Notation

Interval notation is a simplified way of expressing a set of numbers between two endpoints. It is especially useful for denoting the domain of a function.

In this notation, a round bracket \(( )\) indicates that the endpoint is not included in the interval, known as an open interval. A square bracket \([ ]\), on the other hand, means the endpoint is included, called a closed interval.

• For example, the interval \((a, b)\) includes all numbers greater than \(a\) and less than \(b\).
• The interval \([a, b]\) includes all numbers from \(a\) to \(b\), including \(a\) and \(b\) themselves.

In our exercise, the function \(f(x) = \frac{6}{x^2}\) has a domain excluding \(x=0\), leading to the interval notation \(( -\infty, 0) \cup (0, \infty)\). This indicates all real numbers except \(x=0\), or all numbers to the left of zero combined with all numbers to the right.