Final answer:
The question involves matching functions with domain restrictions. The key restriction for these functions is avoiding division by zero. The exact matches for restrictions depend on the definitions of p(x) and q(x).
Step-by-step explanation:
To match each function with the corresponding restrictions to its domain. Each of the given functions has different restrictions based on the operations involved.
- r(x) = p(x)/q(x) - The domain is restricted by the condition that q(x) ≠ 0 to avoid division by zero. Therefore, the restriction could be b) x ≠ -1 or d) x ≠ 0 depending on the definition of q(x).
- r(x) = 1 / (p(x) - q(x)) - Similar to division, we must ensure that p(x) - q(x) ≠ 0 to avoid undefined values. The exact restriction will depend on the functions p(x) and q(x).
- r(x) = 1 / (p(x) + 16q(x)) - Again, the denominator cannot be zero, so we must have p(x) + 16q(x) ≠ 0. The restriction could be any option that ensures the denominator is non-zero.
- r(x) = p(x) ∘ q(x) - Since the composition of functions doesn’t inherently restrict the domain, there are typically 'no restrictions' unless q(x) or p(q(x)) introduces specific restrictions.
- r(x) = (p(x)/q(x)) + 2 - The function is still bound by the restriction that q(x) cannot be zero. This is similar to the first function.
It should be noted that the specific x values where q(x) would become 0 are not provided, so while the principle of avoiding division by zero is clear, the exact matching (a-d) should be determined by the explicit expressions of p(x) and q(x).