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Quadratic functions are a common type of polynomial function with the general form \(f(x) = ax^2 + bx + c\). In this form, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The most distinctive feature of a quadratic function is the \(x^2\) term, which makes the graph of the function a parabolic shape.

• The parabola can open upwards if \(a > 0\), or downwards if \(a < 0\).
• The vertex of the parabola is its highest or lowest point, depending on the direction in which it opens.
• Quadratic functions are symmetrical about a vertical line passing through the vertex.

Quadratic functions are found everywhere in real life, such as in physics for projectile motion or in finance for profit optimization. The function in the exercise, \(g(x) = 1 - 2x^2\), is a simple quadratic equation where \(a = -2\), \(b = 0\), and \(c = 1\). This tells us the parabola opens downward, since \(a\) is negative.

Real numbers include all the numbers that can be found on the number line. This includes:

• Positive numbers (e.g., 1, 2.5, 3/4)
• Negative numbers (e.g., -1, -2.5, -3/4)
• Zero
• Irrational numbers (e.g., \(\pi\), \(\sqrt{2}\))

Real numbers are the most familiar type of numbers we use in everyday life, and they encompass both rational and irrational numbers. In mathematics, real numbers have no imaginary component. When a function has a domain extending over all real numbers, it means the input \(x\) can be any real number without restriction.The concept of real numbers is essential in calculus, algebra, and most areas of mathematics dealing with continuous quantities. For the quadratic function in our exercise, all real numbers are the domain, meaning any real value for \(x\) will yield a valid output for \(g(x)\).

Interval notation is a concise way to express the set of all real numbers between two endpoints. It uses parentheses \(()\) and brackets \([]\) to denote open and closed intervals.

• \(()\) denotes an open interval, meaning the endpoints are not included.
• \([]\) denotes a closed interval, meaning the endpoints are included.

For example, \((2, 5)\) means all numbers greater than 2 and less than 5, whereas \([2, 5]\) includes 2 and 5 themselves. When describing the domain of a function, interval notation efficiently communicates which values are permissible.In the exercise, the domain was determined as all real numbers, represented in interval notation as \((-\infty, +\infty)\). This means that there is no restriction on the values of \(x\), and it can be any real number. The use of \(-\infty\) and \(+\infty\) expresses that the domain extends indefinitely in both directions on the number line.