Problem 8 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The domain of \(-x^4+x^3+9x\) is all real numbers \(\mathbb{R}\).

Step 1: Identify Expression Type

The expression is a polynomial: \(-x^{4}+x^{3}+9x\). Polynomials have the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\).

Step 2: Consider Domain Restrictions

Polynomials are defined for all real numbers because they involve only operations of addition, subtraction, multiplication, and non-negative integer exponents, which are defined everywhere.

Step 3: State the Domain

Since there are no restrictions (like division by zero or square roots of negative numbers), the domain of a polynomial expression is all real numbers, denoted as \(\mathbb{R}\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (often represented as "x") that the function can accept. Not all functions can take every real number as input. For example, functions involving division by zero or square roots of negative numbers have limited domains. In straightforward terms, the domain tells us where the function is "allowed" to work without breaking any mathematical rules.

To find the domain, follow these steps:

• Identify any restrictions, like division by zero or square roots of negative values.
• If there are no operations that could cause issues, the domain is all real numbers, represented as \(\mathbb{R}\).

In the case of a polynomial, like \(-x^4+x^3+9x\), since it only involves multiplication and addition of reals, its domain is all real numbers.

Real Numbers

Real numbers include every number you can think of, except for complex numbers. These consist of integer numbers like -3, 0, 5, and decimal numbers like 1.5 and -2.8. Real numbers form a continuous line without any gaps, representing all the points along a line in geometry. They are denoted by \(\mathbb{R}\).

Why are real numbers important? They help us quantify real-world phenomena and everyday calculations. They form the basis for most of the arithmetic and algebraic structures we deal with in math.

You encounter real numbers daily, such as:

• Counting items: "I have 3 apples" involves integers.
• Reading a thermometer: "It's 23.7 degrees Celsius" involves decimals.

In polynomial expressions, every coefficient and result in a calculation are real numbers.

Polynomial Expression

A polynomial expression is a sum of terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. Polynomials look like this: \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\).

The degree of a polynomial is the highest power of the variable that appears with a non-zero coefficient. In our expression, \(-x^4 + x^3 + 9x\), the degree is 4 because the highest power is \(x^4\).

Some important features of polynomials:

• They do not have variables in denominators or under square roots.
• They are continuous and smooth curves when graphed.
• Their domain is all real numbers, \(\mathbb{R}\), so they are always defined everywhere.

Understanding polynomials is crucial because they model a wide range of real-world problems and scenarios, from physics to economics.