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The domain and range of a function are essential concepts when graphing any mathematical function.
The **domain** refers to all possible input values (x-values) for which the function is defined. For the function \(f(x) = \sqrt[3]{x} + 1\), the domain is all real numbers because you can take the cubic root of any real number. This means the domain is \(-\infty, +\infty\).
The **range** refers to all possible output values (y-values) that the function can produce. Since the cubic root of any number can yield any real result and adding 1 shifts the whole set of results up by 1, the range is also all real numbers. This means the range is \(-\infty, +\infty\).
In a nutshell:

• Domain: \(-\infty, +\infty\)
• Range: \(-\infty, +\infty\)

Understanding the domain and range helps us know the extent of the function's graph horizontally and vertically.

A cubic root function is a type of function that involves the cubic root (or third root) of a variable.
For the given function \(f(x) = \sqrt[3]{x} + 1\), the cubic root symbol \(\sqrt[3]{x}\) indicates that we're looking for a number that, when multiplied by itself three times, yields \(x\).
Characteristics of a cubic root function include:

• It produces both positive and negative results. For example, \(\sqrt[3]{8} = 2\) and \(\sqrt[3]{-8} = -2\).
• Its graph passes through the origin (0,0) if there is no vertical shift.
• It has an 'S' shape curve.

For \(f(x) = \sqrt[3]{x} + 1\), the graph will have a vertical shift, which we'll cover in the next section.

A vertical shift occurs when a constant is added or subtracted from a function.
For our function \(f(x) = \sqrt[3]{x} + 1\), the '+1' indicates a vertical shift.
This means the entire graph of the cubic root function is moved up by 1 unit.
So, while \(\sqrt[3]{x}\) would pass through (0,0), \(\sqrt[3]{x} + 1\) will pass through (0,1).
To graph a vertically shifted cubic root function:

• Identify points on the base cubic root graph, like (-8,-2), (0,0), (8,2).
• Add the vertical shift to the y-values of these points. For example, (-8,-2 + 1) = (-8,-1), (0,0+1) = (0,1), and (8,2 + 1) = (8,3).
• Plot these adjusted points and draw a smooth curve passing through them.

The result is a graph that maintains the 'S' shape but is vertically adjusted by the given constant.