jee-advanced 2024 Q8

jee-advanced · India · paper2 4 marks Composite & Inverse Functions Custom Operation or Property Verification
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that $f ( x + y ) = f ( x ) + f ( y )$ for all $x , y \in \mathbb { R }$, and $g : \mathbb { R } \rightarrow ( 0 , \infty )$ be a function such that $g ( x + y ) = g ( x ) g ( y )$ for all $x , y \in \mathbb { R }$. If $f \left( \frac { - 3 } { 5 } \right) = 12$ and $g \left( \frac { - 1 } { 3 } \right) = 2$, then the value of $\left( f \left( \frac { 1 } { 4 } \right) + g ( - 2 ) - 8 \right) g ( 0 )$ is $\_\_\_\_$ .
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that $f ( x + y ) = f ( x ) + f ( y )$ for all $x , y \in \mathbb { R }$, and $g : \mathbb { R } \rightarrow ( 0 , \infty )$ be a function such that $g ( x + y ) = g ( x ) g ( y )$ for all $x , y \in \mathbb { R }$. If $f \left( \frac { - 3 } { 5 } \right) = 12$ and $g \left( \frac { - 1 } { 3 } \right) = 2$, then the value of $\left( f \left( \frac { 1 } { 4 } \right) + g ( - 2 ) - 8 \right) g ( 0 )$ is $\_\_\_\_$ .