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When working with mathematical functions, the concept of absolute value is crucial; it refers to the distance of a number from zero on the number line, without considering direction. Represented by two vertical bars (| |), it converts negative numbers to positive ones.

For example, in the step function above, we see the term |\frac{1}{8}+4|, which simplifies to |4.125|. The absolute value of 4.125 is 4.125, as it's already a positive number. Similarly, |-4+4| simplifies to |0|, and since 0 is neither positive nor negative, its absolute value remains 0. Understanding absolute value is fundamental when evaluating step functions because these functions often involve it to determine different outcomes based on the input value.

When writing absolute values in Latex, the syntax is \( |x| \), which renders as |x|. In the given exercise, absolute value helps to define the pieces of the function based on the sign of the input.

A piecewise function is a function that is defined by multiple sub-functions, each of which applies to a certain interval of the domain. In the context of the exercise, although the function g(x) = -7[|x+4|]+6 is not explicitly given as a piecewise function, the use of the absolute value creates distinct ‘pieces’ that operate differently depending on whether the input value (x+4) is positive or negative.

Understanding Piecewise Functions

To fully grasp piecewise functions, imagine breaking down a function into parts, where each part has its own rule. As absolute value creates a 'split' at 0, it naturally divides the function into at least two pieces. Students often visualize these functions using a graph because it can clearly show where one part of the function stops and another starts.

When solving problems, identifying ranges of the input for which each piece of the function applies is essential. This becomes evident through practice with diverse functions and different intervals.

The process of function evaluation involves finding the output of a function for a particular input. This is done by substituting the input value into the function formula and simplifying.

In the provided exercise, each evaluation step starts with substituting a different value of x into the function to find the corresponding value of g(x). For instance, g(9) involves replacing x with 9 in the function -7[|x+4|]+6 to obtain -85. The ability to evaluate functions is a fundamental skill in algebra and pre-calculus, and it extends to more complex functions in higher mathematics.

Evaluating functions carefully, especially piecewise and step functions, assures correct solutions even when the function's rule changes with different inputs. Practicing function evaluation with a variety of functions and inputs hones this skill and builds the foundation for more advanced subjects such as calculus.