Injective Function: Cubic Function Real Numbers?

In summary, a cubic function is injective for all real numbers if its derivative never changes sign. However, for a function to have an inverse, it must be bijective, meaning it must be both injective and surjective.
  • #1
jackson
6
0
is a cubic function injective for all real numbers?
 
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  • #2
Yes it's an injective function.
 
  • #3
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.
 
  • #4
Also because to have an inverse function [itex] f^{-1} (x) [/itex] it must be bijective (injective and surjective).
 
  • #5
Well Thx A Great Deal Guys I'm Struggling With First Year Maths And This Site Really Helps, Thx A Million
 

Related to Injective Function: Cubic Function Real Numbers?

1. What is an injective function?

An injective function is a type of mathematical function where each element in the domain maps to a unique element in the range. This means that no two elements in the domain will have the same output value in the range.

2. How is a cubic function different from other types of functions?

A cubic function is a type of polynomial function where the highest degree term is raised to the third power. This makes it different from linear, quadratic, and other types of functions that have different degrees for their highest degree term.

3. Are all cubic functions injective?

No, not all cubic functions are injective. An injective cubic function must have a one-to-one mapping between the domain and range. This means that each input value in the domain must have a unique output value in the range. If this condition is not met, the cubic function is not injective.

4. Can a cubic function have multiple roots?

Yes, a cubic function can have multiple roots. A root is a value that makes the function equal to zero. Since a cubic function is a polynomial function, it can have up to three real roots. However, an injective cubic function can only have one root.

5. How is a cubic function graphed?

A cubic function can be graphed by plotting points on a coordinate plane and connecting them with a smooth curve. The shape of the curve will depend on the coefficients of the function and can be used to determine the behavior of the function.

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