Let's break down the given problem:

The problem asks for values of $a$ and $b$, given that: $f : \mathbb{R} \rightarrow \mathbb{R}$ and the function is defined as: $f(x) = (a + 1) x + b - 2$

The question also states that $f(x)$ relates each real number to itself, meaning that: $f(x) = x$ for all real $x$.

Thus, we set: $(a + 1) x + b - 2 = x$ Simplifying this equation: $(a + 1) x + b - 2 = x$ Subtracting $x$ from both sides: $(a + 1 - 1) x + b - 2 = 0$ which simplifies to: $a x + b - 2 = 0$ For this to hold true for all $x$, we must have: $a = 0 \quad \text{and} \quad b - 2 = 0$

Solving for $b$: $b = 2$

Therefore, the values of $a$ and $b$ are: $a = 0 \quad \text{and} \quad b = 2$

So, the correct answer is $\boxed{(0, 2)}$, which matches option (c).

Would you like further clarification on this problem or have any other questions?

Here are 5 related questions for practice:

1. What is the general solution for a linear equation in one variable?
2. How does one identify the slope and intercept of a linear function?
3. Can a linear function ever map real numbers to themselves under conditions other than identity?
4. How would this problem change if the given equation was quadratic instead of linear?
5. What is the role of the function's domain and range in real analysis?

Tip: When simplifying an equation involving variables, always group terms involving the same variable to isolate and solve for unknowns efficiently.