In Exercises 15-20, sketch the graph of each function and...

In Exercises $15-20,$ 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 $1 .$ $$ h(x)=\left\{\begin{array}{ll}{\frac{1}{x},} & {-1 \leq x<0} \\ {\sqrt{x},} & {0 \leq x \leq 4}\end{array}\right. $$

In Exercises $15-20,$ 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 $1 .$
$$
h(x)=\left\{\begin{array}{ll}{\frac{1}{x},} & {-1 \leq x<0} \\ {\sqrt{x},} & {0 \leq x \leq 4}\end{array}\right.
$$

Key Concepts

Absolute Extreme Values

Absolute extreme values refer to the highest and lowest function values that a function attains over its entire domain. An absolute maximum is the greatest output value, and an absolute minimum is the smallest output value for the function. Finding these values involves evaluating the function at critical points as well as at the endpoints of the domain, keeping in mind that a function may not achieve one or both if it is unbounded or discontinuous at some point in its domain.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed, bounded interval, then it must attain both an absolute maximum and an absolute minimum on that interval. This theorem underscores the importance of both continuity and the nature of the interval (closed and bounded) in guaranteeing the existence of extreme values. When applying this theorem, one must be cautious with piecewise or discontinuous functions since they may not meet the theorem’s conditions, thereby affecting the existence of absolute extrema.

Piecewise-Defined Functions

Piecewise-defined functions are functions whose definitions are split into different cases, each applying to a specific part of the domain. Due to their segmented nature, careful analysis is required at the boundaries between pieces to check for continuity or potential jumps. This is particularly important when examining extreme values, as different rules in different segments can lead to inconsistencies which affect whether or not the function attains an absolute maximum or minimum.

Function Domain

The domain of a function is the set of input values for which the function is defined. When determining extreme values, it is crucial to consider the exact domain of each piece of a piecewise function. This includes identifying whether the interval is closed or open at its endpoints, as the absence of an endpoint in the domain can have significant implications for the existence of absolute extrema.

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