Math function no idea

Question

How do we solve this question?

Consider the three statements:

1. f(a+b)=f(a)+f(b)

2. f(ab)=f(a)f(b)

3. f(1/a)=1/f(a)

What are the statements that are true for every function f?

Answer 1

None are true for EVERY single function f(x). I will give you true and false examples.

(1)

f(x) = kx,

where k is a constant.

f(a + b) = k(a + b) = ka + kb = f(a) + f(b)

Now consider

f(x) = x²

f(a + b) = (a + b)² = a² + 2ab + b² ≠ f(a) + f(b)

(2)

f(z) = z²

f(ab) = (ab)² = a²b² = f(a)f(b)

Now consider

f(x) = x + 1

f(ab) = ab + 1

f(a)f(b) = (a + 1)(b + 1) = ab + a + b + 1 ≠ ab + 1 (unless a+b=0)

f(ab) ≠ f(a)f(b) in this case

Now, can you apply this kind of reasoning and show that it is not generally true that f(1/a) = 1/f(a)?