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.
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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.
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