GMAT Function - This is probably really easy, but I'm not seeing it?

Question

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

a) f(x) = x^2
b) f(x) = x +1
c) f(x) = sqrt x
d)f(x) = 2/x
e)f(x) = -3x

Answer 1

The best way to approach this question is to replace the function in the original equation with the function from each answer and see whether the equation still holds true.

Let's work through them.

a) f(x) = x^2

Replace into the equation

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

If we expand the LHS side of the equation we get

a^2 + 2ab + b^2 = a^2 + b^2

Which you should be able to see is not true.


b) f(x) = x + 1 gives us

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

Which you should be able to see is not true.


c) f(x) = sqrt x gives us

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

Which you should know is not true (replace a and b with 9 and 16 respectively if you are not sure about this)


d) f(x) = 2/x gives us

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

Again this is clearly not true (try replacing a and b with 1 if you are unconvinced)


e) f(x) = -3x gives us

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

Multiplying out the brackets gives us

-3a - 3b = -3a - 3b

Which is clearly true and therefore E is the correct answer.

Answer 2

The property f(a + b) = f(a) + f(b) is a property of linear functions.

The only function in that list that is LINEAR is (e).

(note that some people would say (b) is linear, but (b) is actually affine; it does not have the property you want)

Just test it out. For the function (e):

f(a + b) = -3*(a+b) = -3*a + -3*b

However:

f(a) = -3*a
f(b) = -3*b

Thus,

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

(notice that this does NOT hold for (b) or any of the others)

So (e) is your answer.

Answer 3

a) f(x) = x^2
f(a+b)=(a+b)^2=a^2+2ab+b^2
f(a)+f(b)=a^2+b^2
NO

b) f(x) = x +1
f(a+b)=a+b+1
f(a)+f(b)=a+1+b+1=a+b+2
NO

c) f(x) = sqrt x
f(a+b)=sqrt(a+b)
f(a)+f(b)=sqrt(a)+sqrt(b)
let a=2 & b=3
sqrt(a+b)=sqrt(5)=2.236
sqrt(2)+sqrt(3)=3.146
NO

d)f(x) = 2/x
f(a+b)=2/(a+b)
f(a)+f(b)=2/a+2/b
let a=2 & b=3
f(a+b)=2/5=.4
f(a)+f(b)=2/2+2/3=5/3
NO

e)f(x) = -3x
fIa+b)=-3(a+b)=-3a-3b
f(a)+f(b)=-3a-3b
YES

Only for e

Answer 4

the answer is E

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

distribute

-3a + -3b = -3a + -3b