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When dealing with piecewise-defined functions like \(G(x)\), understanding the domain and range is crucial.

The domain of a function indicates all the possible input values (or \(x\) values) that can be plugged into the function. For \(G(x)\), the domain comes from combining all the permissible \(x\) intervals of each piece.

• The first segment, \(-\sqrt[3]{x}\), allows all \(x\leq-1\).
• The middle segment, \(x\), is defined for \(-1 < x < 1\).
• Lastly, the segment \(-\sqrt{x}\), is valid for \(x > 1\).

Thus, the overall domain is expressed in interval notation:
\((-\infty, -1] \cup (-1, 1) \cup (1, \infty)\).

The range, on the other hand, represents all possible output values of the function. Similar to how the domain was constructed, the range combines the outcomes of the three pieces:

• For \(-\sqrt[3]{x}\), the range is \([1, \infty)\).
• The linear function part, \(x\), gives a range of \((-1, 1)\).
• For \(-\sqrt{x}\), the range is \((-\infty, -1)\).

Putting it all together, the range in interval notation becomes:
\((-\infty, -1) \cup (-1, 1) \cup [1, \infty)\).

Interval notation is a useful way to describe sets of numbers, often used for domains and ranges.

This notation relies on brackets to show whether endpoints are included or not. Important points to consider are:

• Round brackets \(()\) indicate that an endpoint is excluded.
• Square brackets \([]\) mean an endpoint is included.
• The symbol \(\infty\) means boundlessness, and it is always paired with round brackets because infinity can never be reached.

So, for instance, an interval \((a, b] \) includes all numbers greater than \(a\) and up to and including \(b\).
Similarly, an interval like \((-\infty, a)\) represents numbers that are less than \(a\), stretching indefinitely to the left.

In practical use, piecewise functions like \(G(x)\) utilize interval notation to concisely express their domains and ranges. This simplifies understanding which parts of the graph are covered by each segment, and what the full behavior of the function is across the entire meaningful span of \(x\) values.

Piecewise functions can display various behaviors depending on the segment being analyzed.

Understanding how a function increases or decreases over intervals is integral for graphing and analysis. For \(G(x)\), these points are critical:

• In the part defined by \(-\sqrt[3]{x}\) for \(x \leq -1\), the function is decreasing. This means as \(x\) gets more negative, the function's output becomes smaller.
• The middle piece, which is the linear segment \(x\) for \(-1 < x < 1\), increases. Within this interval, outputs increase as \(x\) increases.
• The segment \(-\sqrt{x}\) for \(x > 1\) also decreases. Here, as \(x\) becomes larger, the function yields smaller outputs.

It can help to visualize these changes with a graph to see exactly where the increase or decrease occurs.
Recognizing such trends allows us to predict the behavior of a function over its entire domain, and it provides insight into critical points and transitions from increasing to decreasing behavior.