Functions

Concept:The condition $g(f(x)) = 2^x$ forces $f$ to be injective. Then count injective functions from $A$ to $B$ with restrictions $f(2) \neq 2$ and $f(4) \neq 4$.Explanation:Since $2^x$ is unique for each $x$, $f$ must be one-one. Total one-one functions from $A$ (5 elements) to $B$ (7 elements) = ${}^7C_5 \times 5! = 21 \times 120 = 2520$.Subtract those violating $f(2) \neq 2$ or $f(4) \neq 4$.Cases: $f(2)=2$: choose images for remaining 4 from remaining 6: ${}^6C_4 \times 4! = 15 \times 24 = 360$.Similarly $f(4)=4$: also $360$.Both $f(2)=2$ and $f(4)=4$: choose images for remaining 3 from remaining 5: ${}^5C_3 \times 3! = 10 \times 6 = 60$.By inclusion-exclusion: $2520 - (360+360) + 60 = 1860$.Answer:$1860$