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The absolute value is a key concept in mathematics. It refers to the distance a number is from zero on the number line. This value is always non-negative, because distance cannot be negative. In mathematical terms, the absolute value of a number \( a \), denoted as \( |a| \), can be defined as:

• \( |a| = a \) if \( a \geq 0 \)
• \( |a| = -a \) if \( a < 0 \)

This means if you have a negative value inside the absolute value, you multiply by \(-1\) to make it positive. For example, \(|-5| = 5\) and \(|3| = 3\). When dealing with expressions inside absolute values, always solve the expression first and then apply the absolute condition. For example, \(|x - 2|\) means to subtract 2 from \(x\) first, and then take the absolute value of that result.

Evaluating functions is about finding out what a function equals for specific input values. For the given function \( f(x) = |x-2| + |x+1| - 5 \), our task is to substitute different \( x \) values into the function to solve for \( f(x) \). The steps are simple:

• Substitute the value of \( x \) into the function.
• Calculate the result of each expression inside the absolute value.
• Apply the absolute value rule to any negative results.
• Add or subtract according to the function's operations.

For example, when evaluating \( f(-3) \), substitute \( x = -3 \):
\[ f(-3) = |-3 - 2| + |-3 + 1| - 5 \]
Simplify to get:
\[ = | -5 | + | -2 | - 5 \]
\[ = 5 + 2 - 5 = 2 \]
Each step involves clear computation and application of absolute value properties for accurate results.

A piecewise function is a function composed of several sub-functions, each applying to a particular interval of the main function's domain. It allows a function to behave differently based on the input value. Although our given function isn't explicitly a piecewise function, it's beneficial to understand the concept when dealing with complex absolute value functions, as they can sometimes be broken into cases.In the function \( f(x) = |x-2| + |x+1| - 5 \), breaking down the absolute value expressions can resemble analyzing a piecewise function. For example, for \(|x-2|\), we consider:

• \(x-2 \geq 0\) which simplifies to \(x \geq 2\), yielding \(|x-2| = x-2\)
• \(x-2 < 0\) which simplifies to \(x < 2\), yielding \(|x-2| = 2-x\)

Applying similar logic to \(|x+1|\), the function can be approached as dependent on the value of \(x\). Recognizing these intervals helps in accurately computing the given value, ensuring precise evaluation across different sub-ranges.