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Cubic functions are a type of polynomial function where the highest degree of the variable is three. These functions look like this:

• General form: \(f(x) = ax^3 + bx^2 + cx + d\)
• Specific function we're working with: \(f(x) = x^3 + 1\)

This specific function takes the graph of \(x^3\) and shifts it up by 1 unit. Cubic functions are known for their smooth, continuous curves and can have up to three real roots. They can also display various types of symmetry, sometimes being symmetric about a point called inflection point.
Another interesting property is that cubic functions cover all real numbers. This means for every \(x\), there is a unique \(y\), ensuring the function is reversible, allowing us to find an inverse.

Understanding the domain and range is crucial when working with functions and their inverses. Let's define these terms:

• Domain: All possible input values (\(x\)-values) for which the function is defined.
• Range: All possible output values (\(y\)-values) that the function can produce.

For the function \(f(x) = x^3 + 1\), the domain is all real numbers since you can cube any real number, and adding one doesn't impose restrictions.
The range is also all real numbers because no matter which \(x\) is chosen, \(y\) will cover every possible real number.
The inverse function \(f^{-1}(x) = \sqrt[3]{x - 1}\) also maintains the same domain and range: all real numbers. This symmetry of domain and range across the original and inverse functions is common for cubic and other polynomial functions of odd degree.

Function composition involves applying one function to the results of another. It essentially allows for the combination of two functions into a single operation. When verifying inverse functions, function composition is used to check if one function truly undoes another.

• For a function \(f\) and its inverse \(f^{-1}\), we test: \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

In the given exercise, we verified \(f(f^{-1}(x))\) through:\[f(f^{-1}(x)) = f\left(\sqrt[3]{x - 1}\right) = \left(\sqrt[3]{x - 1}\right)^3 + 1 = x - 1 + 1 = x\]And similarly checked \(f^{-1}(f(x))\) as:\[f^{-1}(f(x)) = f^{-1}(x^3 + 1) = \sqrt[3]{x^3 + 1 - 1} = \sqrt[3]{x^3} = x\]These compositions confirm our inverse function is correct, essentially asserting that \(f\) and \(f^{-1}\) "undo" each other's effects, returning the original input \(x\).