91Ó°ÊÓ

The absolute value is a concept used in mathematics to describe the distance a number is from zero on the number line, without considering direction. It is denoted by vertical bars, for instance, \(|x|\). The absolute value of a number is always non-negative. For example, \(|5| = 5\) and \(|-5| = 5\). The absolute value operation makes any negative number positive, while positive numbers remain unchanged.

Understanding absolute value is crucial when evaluating functions like \(f(x)=\frac{x}{|x|}\). In this function, regardless of whether \(x\) is positive or negative, \(|x|\) always converts it to positive when computing the denominator.

This is why, in our steps to evaluate \(f(5)\) and \(f(-5)\), the denominators \(|5|\) and \(|-5|\) both equal 5. This understanding simplifies the function evaluation by ensuring that only the numerator's sign dictates the result.

The substitution method simplifies the evaluation of functions by replacing variables with given specific values. It's like a plug-and-play for functions, making it easy to see what the function outputs for a given input.

In our exercise, we need to evaluate the function \(f(x) = \frac{x}{|x|}\) for \(x = 5\) and \(x = -5\). By substituting \(x = 5\) into the function, it simplifies to \(f(5) = \frac{5}{|5|} = 1\). Similarly, substituting \(x = -5\) gives us \(f(-5) = \frac{-5}{|-5|} = -1\).

The substitution method is vital because it directly connects the abstract formula of a function to tangible numbers, making the output or result clear and understandable. It's an essential skill in both algebra and calculus, where functions often need to be evaluated at specific points.

A piecewise function is a function defined by multiple sub-functions, each applicable to a certain interval of the main function's domain. Piecewise functions reflect situations where a rule or formula might change based on the input value.

Our function, \(f(x) = \frac{x}{|x|}\), though not explicitly given as piecewise, behaves like one. It effectively changes behavior for positive and negative values of \(x\):

• For positive \(x\), \(f(x) = 1\)
• For negative \(x\), \(f(x) = -1\)

This is because the numerator \(x\) retains its sign, while the denominator \(|x|\) ensures the denominator is always positive.

Piecewise functions are common in mathematics because they can represent real-world situations involving sudden changes, such as tax brackets or speed limits. Understanding them helps in identifying how different rules apply to different input intervals in a single function.