91Ó°ÊÓ

The absolute value of a number is the distance that number is from zero on the number line. It is always non-negative, meaning it's always a positive number or zero. Absolute value is denoted by vertical bars surrounding the number, for example, \(|x|\).
When dealing with the absolute value:

• If the number inside the absolute value is positive or zero, its absolute value is the same as the number itself. For instance, \(|6| = 6\).
• If the number inside the absolute value is negative, you remove the negative sign to make it positive. For example, \(|-6| = 6\).

So, absolute value functions are useful in determining how far away a number is from zero, regardless of its direction on the number line. This property plays a crucial role in simplifying expressions involving absolute values, as seen in our function evaluation.

Substitution is an essential skill in mathematics, often used to replace a variable with a specific number or another expression. In the context of function evaluation, substitution requires plugging in a given number for the variable in the function. For instance, if we have a function \(f(x) = \frac{x}{|x|}\), and we want to evaluate it at a particular value, say 6, we substitute the variable \(x\) with 6. So it becomes \(f(6) = \frac{6}{|6|}\).
Substitution simplifies the original function by turning it into a numerical expression. This method allows you to quickly find out how changes in inputs will affect the outcome, making it particularly useful in a variety of mathematical applications, including real-world problem-solving scenarios.

Positive numbers are all numbers greater than zero. They have significant importance when dealing with function evaluations that include fractions and absolute values.
When evaluating a function at a positive number, calculating the absolute value becomes straightforward. The absolute value of a positive number is simply the number itself, as seen in the expression \(f(6) = \frac{6}{|6|} = \frac{6}{6} = 1\).

• Numbers like 1, 2, 3, and 6 are examples of positive numbers.
• Positive numbers are usually marked without a sign but sometimes with a plus \(+\) sign for clarity.

Understanding positive numbers in the context of absolute values helps in simplifying fractions where the numerator and denominator come from evaluating expressions involving these concepts. This simplification is key to correct and efficient function evaluation.

Negative numbers are those less than zero and hold a special place in mathematics due to their properties in calculations, especially involving inequalities and absolute values.
When working with a function at a negative number, we must take the absolute value to proceed accurately. For a given function like \(f(x) = \frac{x}{|x|}\), substituting \(x = -6\) requires considering the absolute value: \(f(-6) = \frac{-6}{|-6|}\).

• The absolute turns \(-6\) into 6, leading to the computation \(\frac{-6}{6} = -1\).
• This outcome indicates the distinct behavior of negative numbers within fractions since the negative sign affects the final result.

Understanding how negative numbers interact with absolute values allows one to deal confidently with calculations involving these numbers, providing clear insights into their practical applications.