So you test them to see which one (if any) works.

a) f(a+b) = (a+b)^2 = a^2 + 2ab + b^2 but f(a) f(b) = a^2 b^2 which is different.

b) f(a+b) = 1/(a+b) but f(a)f(b) = (1/a) (1/b) = 1/(ab) which is different.

c) f(a+b) = 4(a+b) + 1 = 4a+4b+1 but f(a)f(b) = (4a+1)(4b+1) = 16ab + 4b + 4a + 1 which is different.

c) f(a+b) = 4(a+b) = 4a + 4b but f(a)f(b) = (4a)(4b) = 16ab which is different.

Conclusion: none of the functions satisfy f(a+b) = f(a)f(b).

I´ll let you do the second one.

we want f(a+b) = F(a)*f(b). let's look at each option:

a) f(x) = x^2 then

F(a) = a^2; f(b)= b^2 and f(a+b) = a^2+b^2; and F(a)*f(b) = a^2*b^2 clearly not equal NOPE!

b)If f(x) = 1/x then

F(a+b) = 1/a; F(b) = 1/b. so, F(a+b) = (a+b)/ab and F(a)*f(b) = 1/ab clearly not equal NOPE!

c)If f(x)=4x+1 then

f(a) = 4a+1; f(b) = 4b+1 so, f(a+b) = 4a+4b+2 and F(a)*f(b) = (4a+1)(4b+1) clearly not equal NOPE!

d)If f(x) = 4x then

F(a) = 4a; F(b)= 4b; so, F(a+b) = 4(a+b). and F(a)*f(b) = 4a*4b = 16ab clearly not equal NOPE!