Problem 11 - Elementary Algebra

Question

Find the domain of the equations. Assume that the independent variable is the variable that appears in the expression to the right of the equal sign. $$ y=8 x^{3}-6 $$

Step-by-Step Solution

Answer

Answer: The domain of the given equation is all real numbers, represented as $$x \in \mathbb{R}$$.

Step 1: Write the given equation

The given equation is: $$y = 8x^3 - 6$$

Step 2: Identify the type of equation

This is a polynomial equation, as it has a single variable raised to a non-negative integer power.

Step 3: Consider any restrictions

There are no restrictions for polynomial equations in terms of allowed input value. Polynomial functions can accept any real number as input.

Step 4: Determine the domain

The domain of the given equation is all real numbers, which can be represented as: $$x \in \mathbb{R}$$

Key Concepts

Polynomial Equations

Polynomial equations are mathematical expressions that involve variables raised to whole number powers. They can include constant terms, and their structure is generally quite straightforward. For example, the equation provided, \(y = 8x^3 - 6\), is a polynomial equation. Here, the term \(8x^3\) represents the polynomial part, with \(3\) indicating the power or degree of the polynomial.
One important aspect of polynomial equations is that they can take many forms, such as quadratic, cubic, and quartic, depending on the highest degree of their terms.

• Cubic polynomials like \(8x^3 - 6\) have a degree of 3.
• Quadratic polynomials have a degree of 2.
• Linear polynomials have a degree of 1.

Polynomial equations do not involve negative or fractional powers of the variable, which makes them relatively flexible to work with. When considering the domain, polynomial equations are fairly unrestricted, often able to accept all real numbers as possible input values.

Real Numbers

Real numbers encompass a wide range of values including everything we might commonly think of as numbers. They cover both rational and irrational numbers, which means they can be:

• Rational numbers, like \( \frac{1}{2} \) or \(5.75\), which can be expressed as the quotient of two integers.
• Irrational numbers, such as \( \pi \) or \( \sqrt{2} \), which can't be exactly expressed as fractions.

Real numbers also include integers such as \( ... -3, -2, -1, 0, 1, 2, 3, ... \). Together, they form a continuous number line with no gaps, allowing for a seamless understanding of many mathematical concepts, including the domain of functions.
When it comes to the domain for polynomial equations, real numbers come into play because polynomials, especially those without variables in denominators or under square roots, freely accept all these numbers as inputs. This complete set of real numbers ensures that students have a broad range of values to work with when solving or graphing polynomial functions.

Independent Variable

The independent variable is a key player in functions and equations, acting as the input that influences the output or dependent variable. In our equation \(y = 8x^3 - 6\), the variable \(x\) is independent. It stands to the right of the equal sign, directly determining the value of \(y\) with every change it undergoes.
Understanding the role of an independent variable is crucial for grasping how functions operate. Each input of an independent variable, like \( x \) in a polynomial equation, results in a specific output, allowing us to trace and plot the behavior of the function across a series of values.

• In a graph, the independent variable typically resides on the x-axis, thus controlling the horizontal scaling.
• Changes in the independent variable directly affect the dependent variable, reflecting in the function's slope and intercepts.

When considering a function's domain, it's the scope of values that the independent variable can take that matters. For polynomial functions, like the given \(y = 8x^3 - 6\), the independent variable can span over all real numbers, making it highly versatile for various mathematical explorations.