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.
