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Graphing piecewise functions can seem challenging at first, but once you break it down, it's much simpler. A piecewise function is comprised of several different sub-functions, each applying to a specific part of the domain. Let’s dive into how to graph them step by step:

• Identify each segment of the piecewise function, paying close attention to the conditions attached to each function part because they tell you where each piece applies on the x-axis.
• For every separate condition, plot the respective graph on the coordinate plane. In our example, we have three parts:
  • \(-\sqrt[3]{x}\) for \(x < -1\), creating a curve that increases towards zero.
  • The function \(x\) for \(-1 \leq x < 1\) is simply a line from point \(-1, -1\) to \(1, 1\).
  • Lastly, \(\sqrt{x}\) applies when \(x > 1\), starting just above \(x=1\) and curving upward like half a parabola.
• Mark each endpoint accordingly – an open dot means that point is not included in the graph, while a closed dot indicates it is included.

Working through the above steps will help you visualize how the separate parts fit together, resulting in a complete graph of the piecewise function.

Understanding a function's domain and range is vital for accurately describing its behavior.- **Domain**: This term refers to all the possible input values (x-values) that the function can accept. For the given piecewise function \(G(x)\), each section defined is valid over specific intervals. However, when combined, this function accepts any real value for x, thus its domain in interval notation is \(-\infty, \infty\).- **Range**: The range entails all possible outputs (y-values) a function can produce based on its domain. For the different parts of \(G(x)\), the output values range as follows:

• For \(-\sqrt[3]{x}\), the range is \(-\infty, 0\).
• For \(x\), it is \([-1, 1)\).
• And for \(+\sqrt{x}\), it outputs positive values starting from zero.

Hence, all together, the function's range encompasses all real numbers, represented as \(-\infty, \infty\).Incorporating this knowledge ensures that you can track how inputs transform into outputs, an essential skill for mastering functions.

Interval notation provides a concise way to describe the domain and range of functions. It's a mathematical shorthand that denotes sets of numbers along a continuum.When interpreting or writing interval notation:

• **Round brackets \((, )\)** indicate that an endpoint is not included in the interval (open interval).
• **Square brackets \([, ]\)** signify that an endpoint is included (closed interval).

For example, the domain of \(G(x)\), being \(-\infty, \infty\), uses round brackets because infinity is not a number you can reach, it's a concept. Conversely, if a closed interval, like \([-1, 1)\), had been given, it implies inclusion of \-1\, but not \1\.Remember to list intervals from the lesser value to the greater value, maintaining a logical flow. Mastery of interval notation not only aids in graphing functions effectively but also enhances the understanding of where functions operate.