Problem Solving

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

f(x)= x²
f(x)= x+1
f(x)= √x
f(x)= 2/x
f(x)= -3x

Another approach is to let a = 1 and b = 1 and plug in the values.

So, the question becomes, "Which of the following functions are such that f(1+1) = f(1) + f(1)?"
In other words, for which function does f(2) = f(1) + f(1)?

A) If f(x)=x², does f(2) = f(1) + f(1)?
Plug in to get: 2² = 1² + 1²? (No, doesn't work)
So, it is not the case that f(2) = f(1) + f(1), when f(x)=x²

B) If f(x)=x+1, does f(2) = f(1) + f(1)?
Plug in to get: 2+1 = 1+1 + 1+1? (No, doesn't work)
So, it is not the case that f(2) = f(1) + f(1)
.
.
.
A, B, C and D do not work.
So, at this point, we can conclude that E must be the correct answer.
Let's check E anyway (for "fun")

E) If f(x)=-3x, does f(2) = f(1) + f(1)?
Plugging in 2 and 1 we get: (-3)(2) = (-3)(1) + (-3)(1)
Yes, it works

The correct answer is E

Cheers,
Brent

binaras wrote:Hi

Need help in understanding the solution to the following

Question
For which of the following functions is f(a+b) = f(a) + f(b) for all positive numbers of a & b.

1. f(x) = x squared
2. f(x) = x + 1
3. f(x) = sq root of x
4. f(x) = 2/x
5. f(x) = -3x

The amswer is no.5. Need to understand why 5 is correct and the other answer options are not.

Thanks

Solution:

Choosing convenient numbers is the quickest way to solve this problem. Let's review a bit about functions so we can understand exactly what we are doing when we plug in numbers for a and b.

In short, functions can be graphed in the xy-plane and a function will always have an input (the x-coordinate) and an output (the y-coordinate). As an example, let's use the following function: f(x) = x^2. Let's let x = 2, which is the input. We can plug this into our function to create an output.

f(2) = 2^2 = 4

We see that 2 is the input and 4 is the output. In other words, we have a coordinate pair of (2,4). Now we can utilize this idea as we progress through the question.

The question asks: For which of the following functions is f(a+b) = f(a) + f(b) for all positive numbers a and b? We can rephrase it as:

Is the output of f(a+b) always equal to the sum of the outputs of f(a) and f(b)?

To make this easier, let's plug in some simple values for a and b. Let's say a = 2 and b = 3. Now the question becomes:

Is the output of f(2+3) = f(5) equal to the sum of the individual outputs of f(2) and f(3)?

Our goal is to proceed through the answer choices until we find a function such that f(5) = f(2) + f(3).

A) f(x) = x^2

f(2) = 2^2 = 4

f(3) = 3^2 = 9

f(2+3) = f(5) = 5^2 = 25

We see that 4 + 9 does not equal 25.

Answer choice A is not correct.

B) f(x) = x+1

f(2) = 2 + 1 = 3

f(3) = 3 + 1 = 4

f(5) = 5 + 1 = 6

We see that 3 + 4 does not equal 6.

Answer choice B is not correct.

C) f(x) = √x

f(2) = √2

f(3) = √3

f(5) = √5

We see that √2 + √3 does not equal √5.

If this is hard to see, we know that √2 ≈ 1.4, √3 ≈ 1.7, and √5 ≈ 2.2. Thus 1.4 + 1.7 does not equal 2.2.

Answer choice C is not correct.

D) f(x) = 2/x

f(2) = 2/2 = 1

f(3) = 2/3

f(5) = 2/5

We see that 1 + 2/3 does not equal 2/5.

Answer choice D is not correct.

E) f(x) = -3x

f(2) = -3(2) = -6

f(3) = -3(3) = -9

f(5) = -3(5) = -15

We see that -6 + (-9) does equal -15.

Answer choice E is correct.

Note: Even though this problem specifies f(a + b) = f(a) + f(b), which is a general statement, we know that it will be true for all positive numbers. We chose the specific numbers a = 2 and b = 3 to make the problem's solution easier. The fact that these two numbers worked only for choice E and not the other choices means that the correct choice must be E.

NOTE: this is one of those questions that require us to check/test each answer choice. In these situations, always check the answer choices from E to A, because the correct answer is typically closer to the bottom than to the top.

For more on this strategy, see my article: https://www.gmatprepnow.com/articles/han ... -questions

Cheers,
Brent