sterlinggrey wrote:Can someone explain to me what this even means?

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)=(sq root)x
D. f(x)=2/x
E. f(x)=-3x

There's an algebraic approach to this question, but it takes longer than the approach of plugging in numbers.

To begin, let a=1 and let b=1.

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)?

Now le's check the answer choices.

A) If f(x)=x^2, does f(2) = f(1) + f(1)?
Plugging in 2 and 1 we get: 2^2 = 1^2 + 1^2 (doesn't work)
So, it is not the case that f(2) = f(1) + f(1), when f(x)=x^2

B) If f(x)=x+1, does f(2) = f(1) + f(1)?
Plugging in 2 and 1 we get: 2+1 = 1+1 + 1+1
No, it is not the case that f(2) = f(1) + f(1)
.
.
.
Keep trying each function (none works until we get to E)
.
.
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
If f(x)=-3x, f(2) = f(1) + f(1)

The correct answer is E

Cheers,
Brent

sterlinggrey wrote:Can someone explain to me what this even means?

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)=(sq root)x
D. f(x)=2/x
E. f(x)=-3x

Let's analyze each of the options individually:

1. (a + b)² ≠ (a² + b²) => f(a + b) ≠ f(a) + f(b)
2. (a + b + 1) ≠ (a + 1) + (b + 1) = (a + b + 2) => f(a + b) ≠ f(a) + f(b)
3. √(a + b) ≠ (√a + √b) => f(a + b) ≠ f(a) + f(b)
4. 2/(a + b) ≠ (2/a) + (2/b) = 2(a + b)/ab => f(a + b) ≠ f(a) + f(b)
5. (-3(a + b)) = (-3a) + (-3b) => f(a + b) = f(a) + f(b)

The correct answer is E.