Injective Function: Cubic Function Real Numbers?

[jackson]

is a cubic function injective for all real numbers?

 

[Pyrrhus]

Yes it's an injective function.

 

[mathwonk]

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.

 

[Pyrrhus]

Also because to have an inverse function f^{-1} (x) it must be bijective (injective and surjective).

 

[jackson]

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