Problem Solving

The answer is the last one.. however, I can't figure out how it works.. (there's no explanation on the practice exam)

f(a+b) = f(a) + f(b)
let a = 1 & b = 2

{A} f(x) = x^2
LHS = (1+2)^2 = 9
RHS = (1)^2 + (2)^2 = 1 + 4 = 5
NO

{B} f(x) = x + 1
LHS = 1 + 2 + 1 = 4
RHS = (1+1) + (2+1) = 5
NO

{C} f(X) = sqrt(x)
LHS = sqrt(3)
RHS = sqrt(1) + sqrt(2)
NO

{D} f(x) = 2/x
LHS = 2/3
RHS = 2 + 2/2 = 3
NO

{E} f(x) = -3x
LHS = -3(3) = -9
RHS = -3(1) -3(2) = -9
YES

Answer [spoiler]{E}[/spoiler]

yumi2012 wrote:The answer is the last one.. however, I can't figure out how it works.. (there's no explanation on the practice exam)

f(a+b) = f(a) + f(b)

Option 1: f(x) = x^2
f(a+b) = (a+b)^2
f(a) + f(b) = a^2 + b^2
Eliminate.

Option 2: f(x) = x + 1
f(a+b) = (a+b) + 1
f(a) + f(b) = (a + 1) + (b + 1) = a + b + 2
Eliminate.

Option 3: f(x) = sqrt(x)
f(a+b) = sqrt(a+b)
f(a) + f(b) = sqrt(a) + sqrt(b)
Eliminate.

Option 4: f(x) = 2/x
f(a+b) = 2/(a+b)
f(a) + f(b) = 2/a + 2/b
Eliminate.

Option 5: f(x) = -3x
f(a+b) = -3(a+b)
f(a) + f(b) = (-3a) + (-3b) = -3(a+b)
Select.

Choose E

Cheers

For which of the following functions is f(a+b) = F(a) + f(b) for all positive numbers a and b?

f(x) = X^2
f(x) = X+1
f(x) = square root of x
F(x) = 2/x
F(x) = -3x

Let a=2 and b=3.
Then f(a+b) = f(2+3) = f(5).
Question stem rephrased:

For which of the following functions does f(5) = f(2) + f(3)?

Answer choice A:
f(5) = 5² = 25
f(2) = 2² = 4
f(3) = 3² = 9
25 = 4+9.
Doesn't work.

Answer choice B:
f(5) = 5+1 = 6
f(2) = 2+1 = 3
f(3) = 3+1 = 4
6 = 3+4.
Doesn't work.

Answer choice C:
f(5) = √5
f(2) = √2
f(3) = √3
√5 = √2 + √3.
Doesn't work.

Answer choice D:
f(5) = 2/5
f(2) = 2/2 = 1
f(3) = 2/3
2/5 = 1 + 2/3.
Doesn't work.

The correct answer is E.

Answer choice E:
f(5) = -3*5 = -15
f(2) = -3*2 = -6
f(3) = -3*3 = -9
-15 = -6 + -9.
Success!