Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain of each function using interval notation. \(f(x)=-2 x(x-1)(x-2)\)

Step-by-Step Solution

Answer

The domain of the function is \((- fty, fty)\).

Step 1: Identify Possible Domain Restrictions

The function given is a polynomial function, specifically a cubic polynomial: \(f(x) = -2x(x-1)(x-2)\). Polynomial functions are defined for all real numbers, meaning there are no restrictions like divisions by zero or square roots of negative numbers. Therefore, there are no restrictions to consider for this function.

Step 2: Express the Domain in Interval Notation

Since there are no exclusions in the domain of a polynomial function, the domain includes all real numbers. In interval notation, this is expressed as \((-fty, fty)\).

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental concept in mathematics and are widely used in algebra. But what exactly makes a function a polynomial? Simply put, a polynomial function is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. Notably, these expressions are characterized by non-negative integer exponents on their variables.
For example, a polynomial function could look like this: \(f(x) = -2x^3 + 3x^2 - x + 7\).

• Polynomials do not involve division by a variable.
• They do not have negative or fractional exponents.
• The degree of a polynomial is determined by the highest power of the variable in the expression.

Cubic functions, like the one in the exercise \(f(x) = -2x(x-1)(x-2)\), have a degree of three, meaning their graph may turn direction up to two times, resulting in a curve with up to three zero points.
Importantly, polynomial functions are defined for all real numbers, which leads us to the next topic.

Interval Notation

Interval notation is a concise way of describing the set of all numbers between two endpoints. It is particularly useful in mathematics for specifying the domain or range of a function. Understanding this notation makes it easier to express large groups of numbers without listing them all.
Here's how interval notation works:

• "\([a, b]\)" means all real numbers from \(a\) to \(b\), including both endpoints. The square brackets signify that the endpoints are included in the interval.
• "\((a, b)\)" indicates all numbers between \(a\) and \(b\), excluding the endpoints. Parentheses mean the endpoints are not part of the interval.
• "\((-\infty, \infty)\)" describes all real numbers. Since infinity is a concept rather than a number, it is always written with parentheses.

In the exercise, since polynomial functions such as \(f(x) = -2x(x-1)(x-2)\) are defined for all real numbers, the domain is expressed in interval notation as \((-\infty, \infty)\). This means there are no restrictions on the values that \(x\) can take, covering every possible real number.

Real Numbers

Real numbers are the cornerstone of everyday mathematics. They include all the numbers we typically encounter, such as integers, fractions, and decimals, covering both rational and irrational numbers.
Real numbers can be categorized as follows:

• Integers: Whole numbers, including negative, zero, and positive numbers (e.g., -3, 0, 2).
• Rational numbers: Numbers that can be expressed as a fraction of two integers, where the denominator is not zero (e.g., \(\frac{1}{2}\), 3, -4).
• Irrational numbers: Numbers that cannot be expressed as simple fractions. Their decimal representation goes on forever without repeating (e.g., \(\pi\), \(\sqrt{2}\)).

For polynomial functions, the domain typically includes all real numbers, as they can accept any of these numbers as input. Since polynomial functions do not have division by zero or square roots of negative numbers, they are free from the restrictions that sometimes limit the domain of other types of functions.