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JEE Advance | PYQ

Let \( f:\mathbb{R}\to \mathbb{R} \) be a function such that \( f(x+y)=f(x)+f(y) \) for all \( x,y\in \mathbb{R} \), and \( g:\mathbb{R}\to (0,\infty) \) be a function such that \( g(x+y)=g(x)g(y) \) for all \( x,y\in \mathbb{R} \). If \( f\!\left(\dfrac{-3}{5}\right)=12 \) and \( g\!\left(\dfrac{-1}{3}\right)=2 \), then the value of \( \left( f\!\left(\dfrac{1}{4}\right)+g(-2)-8 \right) g(0) \) is ____________.

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 \( f:\mathbb{R}\to \mathbb{R} \) be a function defined by

Then which of the following statements is TRUE?