Injective Function: Cubic Function Real Numbers?

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A cubic function can be injective depending on its specific form. The function f(x) = x^3 + bx^2 + cx + d is injective if its derivative f'(x) = 3x^2 + 2bx + c does not change sign. For instance, f(x) = x^3 is injective because its derivative is always non-negative. Conversely, f(x) = x^3 - x is not injective as its derivative changes sign. To have an inverse function, a cubic function must be bijective, meaning it must be both injective and surjective.
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is a cubic function injective for all real numbers?
 
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Yes it's an injective function.
 
depends on your definition of a "cubic function".

the function f(x) = x^3 + bx^2 + cx + d, could be considred a cubic function.

it is injective if and only if the derivative f'(x) = 3x^2 + 2bx + c never changes sign.

e.g. f(x) = x^3 is injective because f'(x) = 3x^2 is never negative.

but f(x) = x^3 - x is not injective becuase f'(x) = 3x^2 - 1 is positive for x = 1 and negative for x = 0.
 
Also because to have an inverse function f^{-1} (x) it must be bijective (injective and surjective).
 
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