Problem 16 - Calculus with Concepts in Calculus

Question

Find the domain of the function. $$ f(x)=x^{6}-\sqrt{2} x^{3}-\pi $$

Step-by-Step Solution

Answer

The domain of the function is all real numbers \((- \infty, \infty)\).

Step 1: Understand the Problem

The domain of a function represents all the possible input values (usually 'x' values) for which the function is defined. The function given is a polynomial function.

Step 2: Identify the Type of Function

The function provided is \[f(x) = x^{6} - \sqrt{2} x^{3} - \pi\]This is a polynomial function because it is composed of terms that contain variables raised to whole number exponents.

Step 3: Determine the Domain of a Polynomial Function

For any polynomial function, the domain is all real numbers because the function is defined for every real value of \(x\). There are no restrictions such as division by zero or square roots of negative numbers in this function.

Step 4: Present the Domain

The domain for \[f(x) = x^{6} - \sqrt{2} x^{3} - \pi\]is all real numbers, which can be denoted as \((-\infty, \infty)\).

Key Concepts

Polynomial Function

A polynomial function is an expression made up of variables, coefficients, and non-negative integer exponents. This type of function is important in algebra and calculus due to its well-behaved characteristics and simplicity. The form of a polynomial function is typically:

• The general form is \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are coefficients.
• The highest exponent of the variable \(x\), which is \(n\), is called the degree of the polynomial.
• Each term is called a "monomial", and the degree determines the behavior of the polynomial graph at extreme values of \(x\).

In the original exercise, the function \[f(x) = x^{6} - \sqrt{2} x^{3} - \pi\]is a polynomial because it consists of terms where the variable \(x\) is raised to whole number powers.

Real Numbers

Real numbers consist of all the numbers that can be found on the number line. This includes both rational numbers, such as integers and fractions, and irrational numbers, which cannot be expressed as simple fractions. Real numbers are very important in defining domains of functions because they represent possible values that include:

• Positive numbers like 3 and 5.6.
• Negative numbers like -2 and -3.5.
• Zero, which is neither positive nor negative.
• Irrational numbers like \(\pi\) and \(\sqrt{2}\).

Understanding real numbers is crucial when working on the domain of functions, as seen in the original exercise, where the domain is all real numbers, meaning any real number can be used as an input \(x\) value for the function.

Function Definition

A function is a special relationship where each input is related to exactly one output. Functions can be represented in various forms, such as equations, graphs, or tables. The key characteristics of function definition include:

• Domain: The set of all possible input values (usually 'x' values). For the exercise function \[f(x) = x^{6} - \sqrt{2} x^{3} - \pi\], the domain is all real numbers.
• Codomain: The set of potential outputs that include all possible values the function can take.
• Range: The actual set of outputs the function produces.

Understanding these basic parts helps one in analyzing and working with different types of functions. In our case, the function is a polynomial, making its domain simple and unrestricted—spanning all real numbers.