Question

Could someone please explain the processes for these Algebra II function problems?
1. If f is a function such that f(x+2)=f(x)+f(2) show each of the following.
a. f(0)=0
b. f(-2)=-f(2)
2. If f is a function such that f(-x)=-f(x), show that f(0)=0

Answer 1

f(x+2) = f(x) + f(2)
f(x) = f(x+2) - f(2)
put x = 0
f(0) = f(2) - f(2) = 0 (proves a.)

put x = -2
f(-2) = f(0) - f(2) = 0 - f(2)
f(-2) = -f(2) (proves b.)

Answer 2

2. f(-x) = -f(x)
put x = 0
f(-0) = -f(0)
-0 = 0. So
f(0) = -f(0)
f(0) = -1 * f(0)
The only number that is unchanged when you multiply by -1 is zero.
Therefore, f(0) = 0.