Make a table listing ordered pairs for each function. Then...

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which $f$ is increasing, decreasing, or constant.
$$f(x)=\frac{|x|}{x}$$

Key Concepts

Absolute Value Function Properties

Absolute value functions measure the distance of a number from zero, resulting in non-negative outputs. They often introduce a natural split in the behavior of a function, requiring separate consideration of cases when the variable is positive versus when it is negative.

Piecewise Functions

A piecewise function is one that is defined by different expressions for different parts of its domain. This is particularly relevant when functions include absolute value expressions, as the behavior of the function can change based on conditions (such as x being positive or negative).

Domain and Range

This concept involves identifying all possible input values (domain) for which the function is defined and all resulting output values (range). It is crucial when analyzing functions to determine where the function exists and what values it can produce, especially when fractions or absolute values are involved that might restrict the domain.

Graphing Functions

Graphing is the process of plotting ordered pairs from the function's rule on a coordinate plane, which provides a visual understanding of its behavior. It involves both scatterplot creation from a table of values and sketching the overall pattern of the function, considering asymptotes, discontinuities, and other unique features.

Increasing and Decreasing Behavior

This concept refers to intervals over which the function’s output values consistently rise (increasing) or fall (decreasing) as the input increases. It is important for understanding the function's behavior and is often analyzed by studying the function's derivative or by comparing ordered pairs.

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