For which of the following functions is f(a+b)=f(b)+f(a) for all positive numbers a and b?
A. f(x)=x2 B. f(x)=x+1 C. f(x)=√x D. f(x)=2/x E. f(x)=-3x
A. f(x)=x2 B. f(x)=x+1 C. f(x)=√x D. f(x)=2/x E. f(x)=-3x
One approach is to plug in numbers. Let's let a = 1 and b = 1For 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
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)
.
.
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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
