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The absolute value function is a mathematical concept that gives the magnitude of a number irrespective of its sign. It is denoted by two vertical bars around the number or variable, like this:

• The absolute value of a positive number is the number itself.
• For example, \(|3| = 3\).
• The absolute value of a negative number is its positive counterpart.
• For example, \(|-3| = 3\).

The absolute value is used in various applications where only distance or magnitude is important, such as in calculating the distance of a point from the origin on a number line. When evaluating the function given in the exercise, the usage of absolute value ensures that we only consider positive values in the denominator, which simplifies calculations and avoids undefined expressions.

Rational functions are formed as the ratio of two polynomial functions. They are represented in the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)eq 0\). In this exercise, the function is a special kind of rational function:

• \(f(x) = \frac{1}{|x|}\), where \(P(x) = 1\) and \(Q(x) = |x|\).

Rational functions are fundamental in mathematics because they can model a variety of practical situations involving rates, proportions, or indirect relationships. Understanding their behavior relies heavily on the domains and asymptotic behavior since division by zero is not defined. In this particular exercise, using the absolute value in the denominator ensures that \(Q(x)\) is always non-zero, provided \(|x| eq 0\). This creates a well-defined function wherever \(xeq 0\).

The substitution method is a useful technique in evaluating functions, solving equations, or simplifying mathematical expressions. It involves replacing a variable with a specific value or expression to find a particular result. This is how to effectively apply the substitution method:

• Identify the variable within the function that needs evaluation.
• Carefully substitute the specific value or algebraic expression into the variable’s place in the function.
• Proceed with any further simplification if necessary.

In the given exercise, you repeatedly substitute different values or expressions into the given function \(f(x) = \frac{1}{|x|}\). For instance, when calculating \(f(2)\), replace \(x\) with 2, and simplify to find \(\frac{1}{|2|} = \frac{1}{2}\). This method is straightforward but requires careful application to avoid errors, especially when dealing with negative values or complex expressions.

Piecewise functions are defined by different expressions depending on the input value regions; that means, they "piece" together different formulas to handle distinct parts of a domain. Although the exercise might not directly entail a piecewise function, understanding them complements function evaluation.

• Piecewise functions are expressed as multiple sub-functions, each applying to a slice of the domain.
• They can be written in the format \(f(x) = \begin{cases} expression_1, & \text{if } condition_1 \ expression_2, & \text{if } condition_2 \end{cases} \).

When solving problems involving different definitions under various conditions, piecewise functions allow for flexibility and precision. Tailor each sub-function to suit particular intervals or conditions, handling values like negative inputs or zero differently. Understanding how they function helps one navigate intricate scenarios where one-size-fits-all doesn't apply.