a. Give an example of an odd function which has x = 0 in the domain and which passes through the origin.
b. Use the definition of an odd function to prove that the conditions in part (a) are true for all odd functions.
a) Consider function `f(x) = x^3` . The value x = 0 belongs to the domain of this function (all real numbers) and the function passes through the origin: `f(0) = 0` .
By definition, a function is odd when `f(-x) = -f(x)` for all x in the domain.
For `f(x) = x^3` , this condition is satisfied:
`f(-x) = (-x)^3 = -x^3 = -f(x)` .
b) The conditions in part a are NOT true for all odd functions. It IS true that if the odd function is defined at x = 0, then it has to pass through the origin because the definition of odd function at x = 0 becomes
`f(-0) = f(0) = -f(0)` . This can only be true if `f(0) = 0` .
However, it is possible for an odd function NOT to be defined at x = 0. For example, `f(x) = 1/x` is an odd function because the condition `f(-x) = -f(x)` is satisfied for all x, but x = 0 is not within the domain of this function.