How do you algebraically demostrate that f ( x ) = x 3 − x...

Abram Boyd

Abram Boyd

Answered

2022-06-24

How do you algebraically demostrate that f ( x ) = x 3 x is not a one to one function?
So, I'm trying to prove that the function its not one to one, but I'm stuck:
f ( x 1 ) = f ( x 2 ) x 1 = x 2 then is a one to one
So x 1 3 x 1 = x 2 3 x 2
x 1 3 x 2 3 x 1 + x 2 = 0
( x 1 x 2 ) ( x 1 2 + x 1 x 2 + x 2 2 ) ( x 1 x 2 ) = 0
( x 1 x 2 ) ( x 1 2 + x 1 x 2 + x 2 2 1 ) = 0
Which is true if x 1 = x 2 but how do I go about proving that is not a one to one function?

Answer & Explanation

Nia Molina

Nia Molina

Expert

2022-06-25Added 21 answers

Explanation:
You can assume f to be one to one and show contradictions.
  0 , 1 R such that 0 3 0 = 0 and 1 3 1 = 0 which contradicts f being a one to one function. Hence, f is not one to one.
fabios3

fabios3

Expert

2022-06-26Added 10 answers

Explanation:
This is also true if x 1 2 + x 1 x 2 + x 2 2 1 = 0
With the p-q formula, treating x 1 as the variable, we get
x 1 = x 2 2 + x 2 2 4 x 2 2 + 1 = x 2 2 + 1 3 x 2 2 4
and x 1 = x 2 2 + 1 3 x 2 2 4
Thus, there are solutions when x 1 x 2 and you are not one to one.

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