Give an example of a cubic (degree 3) function that is bijective. Explain why it is bijective. Give an example of a cubic (degree 3) function that is not biject

Question

Give an example of a cubic (degree 3) function that is bijective. Explain why it is bijective. Give an example of a cubic (degree 3) function that is not bijective. Explain why it is not bijective.

Answer 1

http://www.mathsisfun.com/sets/injective-surjective-bijective.html

 

The above link has great information on the meaning of

injective, surjective and bijective function... Bijective functions are

both injective and surjective and basically means there is a perfect "one-to-one

correspondence" between the members of the sets.

Every input value x for function  f(x) has one and only one output value f(x)...

Every output value of f(x) has one and only one corresponding input value x.

 

f(x) = x³

is a bijective cubic function.

For any input value x there is one and only one output value for f(x).

For any output value of f(x) there is only one input value x that results

in that output value.

 

f(x) = 2x³ - 3x² - 3x +2

is NOT a bijective cubic function because more than one input

value "x" can produce the same output value for f(x)

 

f(-1)  = 2(-1) - 3(1) -3(-1) + 2 = -5 + 5 = 0

f(2) = 2(8) - 3(4) - 3(2) + 2 = 16-12-6+2 = 0

Since two different input values x create the same

output value for f(x) this is not a bijective function.  

 

Answer 2

A bijection is a one-to-one function which means that the graph passes both the vertical and horizontal line tests.  Try out some cubic equations in a graphing calculator (like desmos.com/calculator) and see if you can come up with one that passes both and one that doesn't.