In order to find the correct option, let's use the value of x = 2 in each function and calculate their values.
So we have:
[tex]\begin{gathered} f(x)=-3x^2-4 \\ f(2)=-3\cdot2^2-4 \\ f(2)=-3\cdot4-4 \\ f(2)=-12-4 \\ f(2)=-16 \\ \\ g(x)=4(x+3)^2-68 \\ g(2)=4\cdot(2+3)^2-68 \\ g(2)=4\cdot5^2-68 \\ g(2)=4\cdot25-68 \\ g(2)=100-68 \\ g(2)=32 \\ \\ h(x)=3x \\ h(2)=3\cdot2 \\ h(2)=6 \\ \\ j(x)=2x-62 \\ j(2)=2\cdot2-62 \\ j(2)=4-62 \\ j(2)=-58 \end{gathered}[/tex]So the function that is mapped to 32 when x = 2 is g(x), therefore the correct option is the second one.