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 $1 .$ $$ g(x)=\left\{\begin{array}{ll}{-x,} & {0 \leq x<1} \\ {x-1,} & {1 \leq x \leq 2}\end{array}\right. $$

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 $1 .$
$$
g(x)=\left\{\begin{array}{ll}{-x,} & {0 \leq x<1} \\ {x-1,} & {1 \leq x \leq 2}\end{array}\right.
$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain. Understanding how each piece behaves in its respective interval is crucial for correctly graphing the function and analyzing properties such as continuity and extrema.

Graphing Functions

Graphing a function involves plotting its behavior over the domain. When dealing with piecewise functions, it is important to carefully plot each segment according to its definition and ensure that the segments connect (or do not connect) appropriately at the boundaries.

Absolute Extreme Values

Absolute extreme values, or absolute maximum and minimum, are the highest and lowest function values attained on a given domain. Identifying these requires evaluating both endpoints and any critical points within the domain where the function's behavior could change.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed and bounded interval, then it must have both an absolute maximum and an absolute minimum. Explaining how the function adheres to this theorem can involve discussing the function's continuity on its interval and checking its behavior at the boundary points.

Transcript

Please give Ace some feedback