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.
$$f(x)=|x|+1$$

Key Concepts

Absolute Value Function

The absolute value function, denoted by |x|, outputs the non-negative magnitude of x regardless of its sign. When combined with other operations (as in f(x) = |x| + 1), it shifts the graph vertically and creates a distinctive V-shaped curve that is fundamental in understanding how modulus operations impact the behavior of functions.

Table of Values

A table of values lists specific input (x) and output (f(x)) pairs for a function. This systematic organization helps in understanding the function's behavior at select points and serves as a base for plotting its graph. It is especially useful when dealing with functions like the absolute value function, where the behavior on either side of a particular point (x = 0) differs.

Graphing Functions

Graphing functions involves plotting ordered pairs on a coordinate plane to visualize the relationship between input and output. For functions like f(x) = |x| + 1, the graph reveals characteristic features such as symmetry, intercepts, and vertices, which are essential for understanding the overall behavior and properties of the function.

Domain and Range

The domain of a function refers to all possible inputs (x-values) for which the function is defined, while the range refers to all possible outputs (f(x)-values). Establishing the domain and range is crucial in analyzing the scope of a function's behavior. In many cases, especially with transformations such as vertical shifts, determining these sets also provides insights into potential restrictions or extensions of the function.

Monotonicity

Monotonicity describes the intervals over which a function is increasing, decreasing, or constant. An increasing function has outputs that rise with its inputs, while a decreasing function exhibits outputs that fall as inputs increase. Identifying these intervals for any function, including those involving absolute values, aids in understanding the function's overall behavior and its critical points.

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