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Understanding the domain and range of piecewise functions helps us know which input and output values the function can take on. For a piecewise function, the domain is all the possible input values. It is essentially a combination of the domains from each individual piece of the function. In our example, the function is defined for every real number, hence, the domain is \((-\infty, \infty)\).
The range, however, is a bit more complex because it describes all the possible output values the function can produce. It's best to look at the range for each piece of the given function separately:
- The piece \(f(x) = x^2\) for \(x \leq -1\) means the function will output values starting from 1 (at \(x = -1\)) upwards indefinitely, forming the range \[\,[1, \infty)\,\].
- For \(f(x) = x^3\) with \(-1 < x < 1\), it spans output values between just below -1 and just below 1, creating the range \((-1, 1)\).
- The segment \(f(x) = x\) for \(x \geq 1\) results in outputs starting from 1 and going indefinitely upwards, thus, it's range is \[\,[1, \infty)\,\].
When merged together, the full range of the piecewise function is \([-1, \infty)\), considering all overlapping and individual ranges.

Identifying increasing, decreasing, and constant intervals in piecewise functions reveals how the function behaves over different domains. \ These intervals tell us about the changes in function values as the input values move.
Let's see how this applies:

• \(f(x) = x^2\) over \(x \leq -1\) is a decreasing function, as captured by its derivative, \(f'(x) = 2x\). Since \(x < 0\) in this interval, \(f'(x) < 0\), and the function decreases.


• \(f(x) = x^3\) over the interval \(-1 < x < 1\) is increasing. The derivative here, \(f'(x) = 3x^2\), is always positive, meaning that the function consistently grows in this piece.


• Lastly, \(f(x) = x\) for \(x \geq 1\) also shows an increasing pattern, due to its derivative, \(f'(x) = 1\), which is positive and constant.

This gives us the periods of increasing and decreasing behavior for the function: decreasing over \((-\infty, -1]\), and increasing over \((-1, \infty)\). The function does not have any constant intervals within these domains.

Graphing piecewise functions requires plotting each segment of the function separately and then connecting them smoothly. Here's how to approach graphing them:

• For \(f(x) = x^2\) when \(x \leq -1\), plot the left part of a parabolic curve starting at \((-1, 1)\). This curve extends leftwards with a continuous decrease as \(x\) becomes more negative.


• For the piece \(f(x) = x^3\), sketch the segment going smoothly through the origin between \(-1 < x < 1\). This part of the graph resembles an 'S' shape pivoted at the origin due to the cubic nature.


• Finally, for \(f(x) = x\) over \(x \geq 1\), draw a straight line with a slope of 1 starting from the point \(1,1\) and moving diagonally upward.

Be sure to differentiate open and closed endpoints on the graph to capture the exact endpoints of any intervals. This plus the smoothly connected transitions between pieces help in visualizing the entire function and understanding how each piece interacts with its neighbors.