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A piecewise function is a function that is defined by different expressions for different parts of its domain. For each interval or segment of the input values, a different formula or rule applies. This helps accommodate scenarios where a function needs to behave differently based on the input value.

For instance, the absolute value function can be considered a piecewise function because it behaves differently depending on whether the input is positive or negative. The piecewise definition of the absolute value function can be written as:

• If \( x \textless 0\), then \( |x| = -x \)
• If \( x \textgreater = 0\), then \( |x| = x \)

This means, if we input a negative value, we take its opposite (which makes it positive). If we input a zero or positive value, it remains unchanged. It's akin to splitting the function into pieces, each with its own rule.

The absolute value of a number is the non-negative value of that number without regard to its sign. In essence, it represents the distance of a number from zero on a number line.

For any real number \( x \), the absolute value is denoted as \( |x| \). The rules for evaluating absolute values are:

• If \( x \textgreater = 0\), then \( |x| = x \)
• If \( x \textless 0\), then \( |x| = -x \)

Using the above rules, the absolute value of \( -2.5 \) is calculated as follows:
Since \( -2.5 \textless 0 \), we take the opposite of \( -2.5 \), which is \( 2.5 \). Thus, \( |-2.5| = 2.5 \). The absolute value function essentially turns any negative input into its positive counterpart.

Function evaluation involves substituting a given input value into the function and calculating the output. This process helps us determine the function's behavior at specific points.

To evaluate the absolute value function \( f(x) = |x| \) at \( x = -2.5 \), follow these steps:

• Substitute \( -2.5 \) for \( x \): \( f(-2.5) = |-2.5| \)
• Calculate the absolute value of \( -2.5 \): \( |-2.5| = 2.5 \)
• Thus, \( f(-2.5) = 2.5 \)

This method of plugging in the value and simplifying ensures that you correctly determine the output of the function for any given input. Understanding and practicing function evaluation builds a strong foundation for more complex mathematical concepts.