If f: Z-{-1} → Z where f(x) = (x ^ 3 + 1)/(x + 1) and g: Z-Z where g(x) = x ^ 2 - x + 1 then examine whether the functions are equal

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.

Explore whether the functions f(x) = (x^3 + 1)/(x + 1) and g(x) = x^2 - x + 1 are equal across their domains. This problem involves simplifying rational functions, applying the sum of cubes formula, and comparing polynomial expressions. Suitable for advanced high school students.

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