which of the following functions is F(A + B) = F(A) + F(B) for all positive numbers A and B?

Question

A) F(x) = X^2
B) F(x) = X + 1
C) F(x) = square root of X
D) F(x) = 2/X
E) F(x) = -3X

please explain how to get to the answer. THANKS!

Answer 1

A) F(x) = x^2



So F(a) = a^2 and F(b) = b^2

therefore

F(a) + F(b) = a^2 + b^2



F(a + b) = (a + b)^2 = a^2 + 2ab + b^2



So F(a + b) is not equal to F(a) + F(b)



B) F(x) = x + 1



F(a) = a + 1

F(b) = b + 1

therefore

F(a) + F(b) = a + 1 + b + 1 = a + b + 2



F(a + b) = (a + b) + 1 = a + b + 1



So F(a) + F(b) is not equal to F(a + b)



C) F(x) = sqrt(x)



F(a) = sqrt(a)

F(b) = sqrt(b)

so

F(a) + F(b) = sqrt(a) + sqrt(b)



F(a + b) = sqrt(a + b)



So F(a) + F(b) is not equal to F(a + b)



D) F(x) = 2/x



F(a) = 2/a

F(b) = 2/b

so

F(a) + F(b) = (2/a) + (2/b) = (2a + 2b) / ab



F(a + b) = 2/(a + b)



So F(a) + F(b) is not equal to F(a + b)



E) F(x) = -3x



F(a) = -3a

F(b) = -3b



F(a) + F(b) = -3a + (-3)b = -3(a + b)



F(a + b) = -3(a + b)



In this case F(a + b) = F(a) + F(b)



Hope that helps :)

Answer 2

It's basically asking you to find the function where a number substituted in for x give the same output as the sum of two substituted numbers that add up to the original substituted number.



Ok that was horribly said. I'll try to clarify. If an equation met the criteria, the following would be true:



f(5) = f(3) + f(2)



Because 3 + 2 is 5.



So you could try this with all the equations, but I can tell you the answer is E because E is direct linear. Using the original variables of the problem, -3(a + b) would equal -3(a) + -3(b). This is the distributive property.



Therefore, f(x) = -3x would meet the criteria.

Answer 3

A) No. F(1 + 2) = F(3) = 3^2 = 9, but F(1) + F(2) = 1^2 + 2^2 = 5.



B) No. F(1 + 2) = F(3) = 3 + 1 = 4, but F(1) + F(2) = (1 + 1) + (2 + 1) = 5.



C) No. F(9 + 4) = F(13) = sqrt(13) ~ 3.6, but F(9) + F(4) =

sqrt(9) + sqrt(4) = 3 + 2 = 5.



D) No. F(1 + 3) = F(4) = 2/4 = 1/2, but F(1) + F(3) = 2/1 + 2/3 = 8/3.



E) Yes. F(A + B) = -3(A + B) = -3A - 3B = -3A + (-3B) = F(A) + F(B).

Answer 4

A) No.

e.g. F(1+2) = 3^2 = 9, whereas F(1) + F(2) = 1 + 4 = 5



B) No.

e.g. F(1 + 2) = 3+1 = 4, whereas F(1) + F(2) = 1+1 + 2+1 = 5



C) No.

e.g. F(2+2) = sqrt(4) = 2, whereas F(2) + F(2) = 2*sqrt(2)



D) No.

e.g. F(2+2) = 2/4 = 1/2, whereas F(2) + F(2) = 1+1 = 2



E. Yes

F(A+B) = -3(A+B) = -3A + -3B = F(A) + F(B) for any A, B

Answer 5

The answer is (E). (side note: this is one of the requirements for a function to be a subspace). Let me show you.



A) F(x) = x^2, so

F(A + B) = (A + B)^2 = A^2 + 2AB + B^2

F(A) + F(B) = A^2 + B^2



And as you can see, they're not equal, i.e.

F(A + B) =/= F(A) + F(B)



B) F(x) = x + 1

F(A + B) = A + B + 1

F(A) + F(B) = (A + 1) + (B + 1) = A + B + 2



So they're not equal, i.e.

F(A + B) =/= F(A) + F(B)



C) F(x) = sqrt(x)



F(A + B) = sqrt(A + B)

F(A) + F(B) = sqrt(A) + sqrt(B)



In no way do we have methods of combining the sum of radicals, so they are not equal.



D) F(x) = 2/x

F(A + B) = 2/(A + B)

F(A) + F(B) = (2/A) + (2/B) = (2B + 2A)/(AB)



Not equal.



E) F(x) = -3x

F(A + B) = -3(A + B)

F(A) + F(B) = -3A + (-3B) = (-3)(A + B)



And as you can see, they are clearly equal.

Answer 6

E) F(x) = - 3x



F(A+B) = -3(A+B) = - 3A + -3B = F(A) + F(B)



but for any other function this is not true.



ex: take f(x) = x + 1



then f(a + b) = a + b + 1



but f(a) + f(b) = a+1 + b+1 = a+b +2 (not equal)