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Let \( f : \mathbb{R} \to \mathbb{R} \) and \( g : \mathbb{R} \to \mathbb{R} \) be functions defined by

Let \( a, b, c, d \in \mathbb{R} \). Define the function \( h : \mathbb{R} \to \mathbb{R} \) by
\( h(x) = a\,f(x) + b\!\left(g(x) + g\!\left(\dfrac{1}{2} - x\right)\right) + c(x - g(x)) + d\,g(x),\ x \in \mathbb{R}. \)
Match each entry in List-I to the correct entry in List-II.

The correct option is:

Let \( \mathbb{R} \) denote the set of all real numbers. Consider the polynomial function \( f:\mathbb{R}\to\mathbb{R} \) defined by
\( f(x)=\dfrac{d^{10}}{dx^{10}}\!\left((x^{2}-1)^{10}\right) \), for all \( x\in\mathbb{R} \).
Here \( \dfrac{d^{10}}{dx^{10}}\!\left((x^{2}-1)^{10}\right) \) is the \( 10^{\text{th}} \) order derivative of the function \( (x^{2}-1)^{10} \).
Then which of the following statements is (are) TRUE?

Let \( \mathbb{N} \) denote the set of all natural numbers, and \( \mathbb{Z} \) denote the set of all integers. Consider the functions \( f:\mathbb{N}\to \mathbb{Z} \) and \( g:\mathbb{Z}\to \mathbb{N} \) defined by
Define \( (g^\circ f)(n)=g(f(n)) \) for all \( n\in \mathbb{N} \), and \( (f^\circ g)(n)=f(g(n)) \) for all \( n\in \mathbb{Z} \).
Then which of the following statements is(are) TRUE?