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A quadratic equation is a mathematical expression that involves a variable raised to the second power. It generally takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the exercise, the quadratic equation is given as \( y = x^2 - 6 \). Here, \( a = 1 \), \( b = 0 \), and \( c = -6 \). This means there's a squared term \( x^2 \), but no linear \( x \) term, simplifying the expression. Quadratic equations always form a symmetrical curve called a parabola. This particular parabola opens upwards because the x-squared term has a positive coefficient. The lowest point of this curve is called the vertex, which in this equation is at \( y = -6 \). Understanding the nature of quadratic equations is crucial for analyzing their domains and ranges, as well as their graph shapes.

A function is defined as a relation where each element of the domain (input) relates to exactly one element of the range (output). To determine whether a relation is a function, we can use the "Vertical Line Test" on its graph. If a vertical line only touches one point on the graph at any input value \( x \), then it is a function.In our case, we have the relation \( y = x^2 - 6 \). For any number we plug in for \( x \), we get one and only one value for \( y \). This fulfills the requirement for a function. Therefore, this relation describes \( y \) as a function of \( x \).Functions are powerful because they make it possible to predict the output when any valid input is known, ensuring a predictable relationship between variables.

Real numbers are a vast set of numbers comprising both rational and irrational numbers. They include all numbers you can think of except imaginary numbers. Real numbers can be positive, negative, or zero.In this exercise, when determining the domain, we discussed that \( y = x^2 - 6 \) can take any real number as input. Therefore, its domain includes all real numbers, which we symbolize as \((-\infty, +\infty)\) or \( \{x \mid x \in \mathbb{R}\} \). This unrestricted domain stems from the fact that any real number squared is still a real number, and so the operation \( x^2 - 6 \) always yields a real result.Understanding real numbers is essential in mathematics, as they form the building blocks for constructing and analyzing most algebraic functions.