Problem 6 - Precalculus Mathematics for Calculus

Question

Find the domain of the expression. $$-x^{4}+x^{3}+9 x$$

Step-by-Step Solution

Answer

The domain is all real numbers: \((- \infty, \infty)\).

Step 1: Identify the Expression Type

The expression given is \[-x^4 + x^3 + 9x\]. This is a polynomial expression. Polynomials are functions that are defined for all real numbers unless specified otherwise.

Step 2: Domain of Polynomial Functions

Polynomial functions, such as the one given, have a domain that consists of all real numbers, \((-\infty, \infty)\). This is because there are no restrictions like division by zero or square roots of negative numbers.

Key Concepts

Polynomial Expression

A polynomial expression is a mathematical phrase involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable, say \(x\), is given by:

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

where \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(n\) is a non-negative integer representing the degree of the polynomial.
In our original exercise,

• The polynomial expression is \(-x^4 + x^3 + 9x\).

In this form,

• \(-1x^4\) indicates a term with degree 4,
• \(1x^3\) indicates a term with degree 3,
• \(9x = 9x^1\) is a term with degree 1.

The degree of the entire polynomial is determined by the highest power of \(x\), which in this case is 4.

Real Numbers

Real numbers encompass all the numbers that we encounter in daily life, including

• natural numbers (1, 2, 3, ...),
• whole numbers (0, 1, 2, 3, ...),
• integers (..., -2, -1, 0, 1, 2, ...),
• rational numbers (fractions and decimals that terminate or repeat),
• and irrational numbers (numbers that cannot be expressed as a fraction, for example, \(\pi\) and \(\sqrt{2}\)).

Real numbers can be visually represented on a number line that stretches from negative infinity to positive infinity.

In the context of a polynomial expression, the real numbers are important because they define the set of possible input values, or the domain, for these expressions.

Function Domain

The domain of a function is the complete set of possible values of the independent variable. When dealing with polynomials, it’s particularly straightforward.
A polynomial function, like the one presented earlier \(-x^4 + x^3 + 9x\), is defined for all real numbers since there are no variables in the denominator and no variables under a square root. Thus, no restrictions are present that would otherwise "break" or result in undefined values.

• This means the domain is all real numbers, which we express as \((-\infty, \infty)\).

When you interpret a function's domain, you're essentially asking, "What values can I safely put in for \(x\) and still have a meaningful result?"

Since any real number can substitute \(x\) without causing any mathematical breakdown in the polynomial expression, it highlights their function-friendly nature.