To determine which rational expression has the domain restrictions $x = -6$ and $x = 1$, we need to understand that domain restrictions in rational expressions occur where the denominator is equal to zero. So, let's look for a denominator that would be zero at $x = -6$ and $x = 1$.

Process:

1. Set up the denominator: The restrictions occur where the denominator is zero, so the denominator should factor to something like $(x + 6)(x - 1)$.

2. Form the rational expression: The rational expression could be any function that has this denominator. For example: $\frac{\text{some numerator}}{(x + 6)(x - 1)}$

Example Rational Expressions:

• $\frac{1}{(x + 6)(x - 1)}$
• $\frac{x}{(x + 6)(x - 1)}$
• $\frac{x^2 + 2}{(x + 6)(x - 1)}$

All of these expressions will have domain restrictions at $x = -6$ and $x = 1$, because the denominator becomes zero at those values of $x$.

So, any rational expression of the form: $\frac{\text{numerator}}{(x + 6)(x - 1)}$ would have the domain restrictions $x = -6$ and $x = 1$.

Do you want more details or have any questions?

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Here are five related questions:

1. How do you find the domain of a rational expression?
2. What happens if a value makes both the numerator and denominator zero?
3. Can a rational expression have more than two domain restrictions?
4. How do you simplify a rational expression with complex domain restrictions?
5. What are the implications of domain restrictions in real-world problems?

Tip: Always factor the denominator of a rational expression completely to easily identify the domain restrictions.