Final answer:
An odd function must satisfy f(0)=0 as it meets the condition f(x) = -f(-x), which implies symmetry about the origin and dictates that the function's values at opposite points around the origin are negatives of each other.
Step-by-step explanation:
To show that for an odd function f, we must have f(0)=0, we need to use the definition of an odd function. An odd function satisfies the condition f(x) = -f(-x) for all x in its domain. So if we apply this property to x=0, we get f(0) = -f(-0), which simplifies to f(0) = -f(0). Since the only number that is equal to its own negative is zero, this implies f(0) = 0.
The definition of an odd function also implies symmetry about the origin, which means that for every point (x, y) on the function, there is a corresponding point (-x, -y). For the table to represent an odd function, for every value f(1) there must be a corresponding value f(-1) = -f(1), and for each f(2), there should be f(-2) = -f(2), and so on for all x values provided in the table.