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The absolute value of a number refers to the non-negative value of that number without regard to its sign. In other words, it's the distance of a number from zero on a number line, which is always positive. For example, the absolute value of both \(3\) and \( -3\) is \(3\).

The absolute value is represented by two vertical bars on either side of the number or expression, like \( |x| \). In the context of functions, absolute value can affect the domain since it alters the set of possible inputs. If an absolute value expression is in the denominator, as in \( f(x) = \frac{1}{|x+3|} \), the arguments causing the expression inside the absolute value to be zero will be critical in determining the domain because they will result in a division by zero.

A function becomes undefined when it involves an operation that cannot produce a result within the set of real numbers. A common occurrence of an undefined function is division by zero. Since division by zero is not allowed in mathematics, any input leading to this situation must be excluded from the domain of the function.

In the example \( f(x) = \frac{1}{|x+3|} \), finding where the function is undefined involves looking for values of x that make the denominator equal to zero. For this function, when \( x = -3\), the absolute value \( |x+3| \)= \( |0| \)= \( 0\), which would result in division by zero. To ensure the function remains defined, these specific values are removed from the domain, helping secure a valid domain for the function in question.

Real numbers encompass all the numbers that can be found on the number line, including both positive and negative integers, fractions, and irrational numbers. They are called 'real' because they represent quantities that can exist in reality.

In locating the domain of \( f(x) \), we are searching for all the possible real number inputs x can take. Typically, unless there is a constraint like an undefined expression or a square root of a negative number -- which are not real -- the domain of a function will consist of all real numbers. For our function \( f(x) = \frac{1}{|x+3|} \), apart from \( x = -3\), which we exclude to prevent an undefined expression, the domain includes every other real number, expressing the wide range of inputs the function can accept.