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Absolute value functions, like the one given in the exercise, measure the distance of a number from zero on the number line. In mathematical terms, the absolute value of any number is always non-negative. For any real number \(x\), the absolute value is represented as \(|x|\). This operation removes any negative sign in front of a number, making it positive.

For example, \(|-5| = 5\) and \(|3| = 3\). The absolute value function \(g(x) = |x+7|\) simply translates the number \(x\) by adding 7, and then takes the absolute value of the result.

This makes absolute value functions interesting because they are defined for all real numbers, meaning they don't have any gaps or restrictions in their domain.

Real numbers are the set of numbers that include all the numbers you could potentially think of on a number line. This set includes:

• Natural numbers (1, 2, 3, ...)
• Whole numbers (0, 1, 2, 3, ...)
• Integers (..., -3, -2, -1, 0, 1, 2, 3, ...)
• Rational numbers (such as 1/2, 0.75, which can be expressed as fractions)
• Irrational numbers (numbers like \(\sqrt{2}\) or \(\pi\))

Real numbers are crucial because most mathematical operations, like those involving absolute values, are defined within this set. The domain of absolute value functions generally covers all real numbers, since these functions are well-behaved across the entire number line without becoming undefined.

Interval notation is a shorthand used to describe a range of numbers, or the domain, along a number line without listing every number individually. This notation uses parentheses \(()\) and brackets \([\ ]\). Parentheses indicate that a number is not included in the interval, whereas brackets indicate inclusion.

For example, the interval \( (3, 7] \) includes numbers greater than 3 but not 3 itself, up to and including 7.

When we express the domain of \(g(x) = |x+7|\), we use interval notation as \( (-\infty, \infty) \). This indicates that the function accepts all real numbers from negative infinity to positive infinity, without any exclusions.