To examine whether the functions $f$ and $g$ are equal, we need to compare $f(x)$ and $g(x)$ for all $x$ in the domain where both functions are defined.

Given: $f: \mathbb{Z} \setminus \{-1\} \to \mathbb{Z}, \quad f(x) = \frac{x^3 + 1}{x + 1}$ $g: \mathbb{Z} \to \mathbb{Z}, \quad g(x) = x^2 - x + 1$

First, let's simplify $f(x)$.

Simplifying $f(x)$

The function $f(x)$ is defined as: $f(x) = \frac{x^3 + 1}{x + 1}$

We can use polynomial long division or factorization to simplify this. Notice that $x^3 + 1$ can be factored using the sum of cubes: $x^3 + 1 = (x + 1)(x^2 - x + 1)$

Thus: $f(x) = \frac{(x + 1)(x^2 - x + 1)}{x + 1}$

For $x \neq -1$: $f(x) = x^2 - x + 1$

Comparing $f(x)$ and $g(x)$

Now, let’s compare $f(x)$ and $g(x)$ for $x \neq -1$.

Given: $g(x) = x^2 - x + 1$

We have: $f(x) = x^2 - x + 1 \quad \text{for } x \neq -1$

Since $f(x) = g(x)$ for all $x$ in $\mathbb{Z} \setminus \{-1\}$, the functions are equal on their common domain.

Therefore, we conclude that: $f(x) = g(x)$

Summary

The functions $f$ and $g$ are equal for all integers $x$ except $x = -1$. Since $f$ is not defined at $x = -1$, we examine the functions over the domain where both are defined and find that they are identical.

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Would you like further details or have any questions?

Here are some related questions that might interest you:

1. What is polynomial long division, and how is it performed?
2. Can you provide another example of simplifying a rational function?
3. How do you determine the domain of a function?
4. What is the sum of cubes formula, and how is it derived?
5. Can you explain the differences between the range and the codomain of a function?
6. How can we find points of discontinuity in a rational function?
7. What are some common methods for factoring polynomials?
8. How do we test the equality of two functions analytically?

Tip: When simplifying rational functions, always check for common factors in the numerator and denominator to simplify the expression.