Problem 10 - 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. $$ y=x^{2}+2 x-9 $$

Step-by-Step Solution

Answer

Answer: The domain of a quadratic function in the form y = ax^2 + bx + c is the set of all real numbers, represented as (-∞, ∞).

Step 1: Identify the equation type

The given equation is a quadratic function in the form of: $$ y = ax^2 + bx + c $$

Step 2: Determine if there are any restrictions on the independent variable

For a quadratic function, there are no restrictions on the independent variable (x) because the equation is defined for all real numbers.

Step 3: Write the domain of the equation

Since there are no restrictions on the independent variable (x) and the equation is defined for all real numbers, the domain of the equation is: $$ Domain: (-\infty, \infty) $$

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x with a non-zero coefficient of the highest degree term. Its standard form is \(y = ax^2 + bx + c\), where \(aeq0\), and a, b, and c are constants. Quadratic equations are fundamental in algebra and they exhibit a characteristic parabolic graph when plotted.

One interesting property of quadratics is their symmetry, with a maximum or minimum point known as the vertex. With regards to their domain, quadratic equations are quite inclusive; they are defined over all real numbers, meaning that you can substitute any real number in for the independent variable and compute a corresponding value of y. This makes the domain of a quadratic equation quite straightforward: it is always \( (-\infty, \infty) \) which is the set of all real numbers.

Independent Variable

In the context of a function like \(y = x^2 + 2x - 9\), the term independent variable refers to the input of the function, which is x in this case. The independent variable is the value you can freely choose, and upon which the output (the dependent variable, y) is based. Since functions are a way to express a relationship between variables, the independent variable can be viewed as the cause while the dependent variable is the effect.

An important aspect of the independent variable is that it can take on any value within the domain of the function. For quadratic functions, this means any real number. Understanding the concept of the independent variable is crucial since it influences the shape and position of the curve representing the function in a coordinate plane.

Real Numbers

The set of real numbers is inclusive of rational numbers—integers, fractions, and decimals (both finite and recurring)—as well as irrational numbers, numbers that cannot be precisely expressed as a simple fraction or decimal, like \(\pi\) or \(\sqrt{2}\).

Real numbers are critical when discussing domains because the domain consists of all possible input values that the independent variable can take. For many functions, particularly polynomials and especially quadratics, the domain is the set of all real numbers. The concept of real numbers is grounded in the number line, where every point represents a real number, allowing us to visualize operations and relationships of these numbers.