3 questions from 2025
Let \( \mathbb{R} \) denote the set of all real numbers. Let \( a_i, b_i \in \mathbb{R} \) for \( i \in \{1,2,3\} \). Define the functions \( f : \mathbb{R} \to \mathbb{R},\ g : \mathbb{R} \to \mathbb{R} \) and \( h : \mathbb{R} \to \mathbb{R} \) by
\( f(x)=a_1+10x+a_2x^2+a_3x^3+x^4 \), \( g(x)=b_1+3x+b_2x^2+b_3x^3+x^4 \), \( h(x)=f(x+1)-g(x+2) \).
If \( f(x)\neq g(x) \) for every \( x\in\mathbb{R} \), then the coefficient of \( x^3 \) in \( h(x) \) is:
Let \( \mathbb{R} \) denote the set of all real numbers. Let \( f:\mathbb{R}\to\mathbb{R} \) and \( g:\mathbb{R}\to(0,4) \) be functions defined by \( f(x)=\log_e\!\left(x^2+2x+4\right) \), and \( g(x)=\dfrac{4}{1+e^{-2x}} \). Define the composite function \( f\circ g^{-1} \) by \( (f\circ g^{-1})(x)=f\!\left(g^{-1}(x)\right) \), where \( g^{-1} \) is the inverse of the function \( g \).
Then the value of the derivative of the composite function \( f\circ g^{-1} \) at \( x=2 \) is ___________.
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?
