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
One approach is to plug in numbers. Let's let a = 1 and 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)?
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
