Can an odd function be injective?

For instance, consider a simple odd function f(x) = x^3. This function is both odd, as f(-x) = -(-x)^3 = x^3 = -f(x), and injective, as every unique value of x corresponds to a unique value of f(x).

In the realm of mathematical functions, an odd function exhibits a unique property that distinguishes it from other functions. While it is possible for an odd function to possess the quality of being injective, or one-to-one, this is not always the case.

However, there exist odd functions that are not injective. One such example could be a modified version of the absolute value function, where f(x) is defined as -x for negative x and x for non-negative x, but with an added twist to ensure it's odd (note: this is a conceptual example as it violates the traditional definition of an absolute value function).

An odd function, by definition, satisfies the condition f(-x) = -f(x) for all values of x within its domain. This symmetry around the origin gives odd functions a distinctive graphical appearance, with their graphs often resembling mirror images across the y-axis.

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