Sketch the graph of each function and determine whether the...

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
$$g(x)=\left\{\begin{array}{ll}
-x, & 0 \leq x<1 \\
x-1, & 1 \leq x \leq 2
\end{array}\right.$$

Key Concepts

Piecewise Functions

A piecewise function is defined by different formulas over different parts of its domain. When analyzing such functions, it is essential to study the behavior on each subinterval and examine the transition points where the definition changes, as these points may involve discontinuities or changes in slope that are critical to understanding the overall behavior of the function.

Absolute Extreme Values

Absolute extreme values are the highest or lowest values that a function takes on over its entire domain. To determine these values, one typically evaluates the function at the endpoints of the domain and at any critical points within the intervals, keeping in mind that discontinuities or gaps in the function's definition may result in the non-attainment of a potential minimum or maximum, even if a bound exists.

Continuity

Continuity refers to the property that small changes in the input of a function result in small changes in the output, meaning the graph of the function is unbroken. For many theorems in calculus, such as the Extreme Value Theorem, continuity over a closed interval is a key condition to ensure the existence of absolute extrema. In cases where a function is not continuous, the expected extreme values may not be attained even if bounds exist.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed and bounded interval, then it must attain both an absolute maximum and an absolute minimum on that interval. This theorem underscores the importance of continuity: for functions that are not continuous over their domain, one cannot guarantee the existence of both absolute extrema, which affects how the function is analyzed.

Graphical Analysis

Graphical analysis involves sketching the function to visually inspect its behavior, such as increasing or decreasing trends, discontinuities, and potential extreme values. By analyzing the graph, one can identify regions where the function may fail to reach a bound, which can provide insight into the results obtained by applying theoretical concepts like the Extreme Value Theorem.

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