Problem 3 - Precalculus : Building Concepts and Connections

Question

What is the domain of the function \(f(x)=x^{2}-3 x ?\)

Step-by-Step Solution

Answer

The domain of the function \(f(x) = x^{2} - 3x\) is all real numbers, or in interval notation, \((- \infty, + \infty)\).

Step 1: Identify Type of Function

Firstly, it is important to identify that the function \(f(x)=x^{2}-3x\) is a polynomial function. Polynomial functions are defined for all real numbers.

Step 2: Determine the Domain

Since the function is a polynomial, the domain of this function is all real numbers. In interval notation, this is expressed as \((- \infty, + \infty)\), denoting that the function includes all values from negative infinity to positive infinity.

Key Concepts

Polynomial Functions

A polynomial function is a type of mathematical expression consisting of variables and coefficients. It includes terms in the form of powers of the variable, generally written as follows:

• Constant term: a number on its own.
• Linear term: a variable raised to the power of one, such as \(x\).
• Quadratic term: a variable squared, such as \(x^2\).
• Cubic term: a variable cubed, such as \(x^3\).

Polynomials can have multiple terms, such as \(ax^n + bx^{n-1} + ... + c\), where \(a, b, c\) are constants. These terms are added together to form the polynomial.

In the exercise given, the function \(f(x) = x^2 - 3x\) includes a linear term \(-3x\) and a quadratic term \(x^2\). Because it is in polynomial form, it has no restrictions like divisions by zero or square roots of negative numbers, allowing it to be defined for all real numbers.

Real Numbers

Real numbers consist of a vast set of numbers that can be found on a number line. This includes both rational and irrational numbers, making the category very broad.

• Rational numbers: numbers that can be expressed as a fraction, like \( \frac{1}{2} \) or 4.
• Irrational numbers: numbers that cannot be expressed as a simple fraction, such as \(\sqrt{2}\) or \( \pi \).

Real numbers include all integer numbers, as well as decimals and fractions.

They are very crucial in defining domains for functions because they encompass all possible values the polynomial can take. When we say a polynomial function can be defined for all real numbers, this means that every point on the number line could plug into our function without causing any mathematical dilemmas.

Interval Notation

Interval notation is a mathematical method used to represent a range of numbers. It is especially useful in expressing the domain and range of functions in a compact form.

The notations use brackets and parentheses to indicate whether endpoints are included or not. Here's how the notation works:

• Parentheses \((a,b)\): both endpoints \(a\) and \(b\) are not included.
• Brackets \([a,b]\): both endpoints \(a\) and \(b\) are included.
• Mixed \((a,b]\) or \([a,b)\): one endpoint is included while the other is not.

In the context of polynomial functions, which are defined for all real numbers, the domain is often written in interval notation as \((-\infty, +\infty)\).

This expression indicates that every real number between negative infinity and positive infinity is included, matching the fact that polynomial functions can take any real number as their input.