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November 30th, 2016, 07:50 AM  #1 
Senior Member Joined: Nov 2015 From: United States of America Posts: 167 Thanks: 21 Math Focus: Calculus and Physics  Best way to show that a function is onetoone
Hello. I am trying to find out what would be the best way to show that a function is onetoone without graphing and doing the horizontal line test. Would I just perform a concavity test for all values on a designated interval? For example, say I have f(x) = x^3 + 2x + 1 . How could I show that this function is or is not onetoone? Many thanks Jacob 
November 30th, 2016, 08:35 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,755 Thanks: 1405 
$f(x) = x^3 + 2x + 1$ $f'(x) = 3x^2 + 2 > 0 \text{ for all } x \in \mathbb{R} \implies f(x) \text{ is strictly increasing}$ ... how would this help to show $f(x)$ is 11? Look at the definition of an injective function. 
November 30th, 2016, 08:47 AM  #3  
Senior Member Joined: Nov 2015 From: United States of America Posts: 167 Thanks: 21 Math Focus: Calculus and Physics  Quote:
Appreciate it.  
November 30th, 2016, 08:55 AM  #4  
Math Team Joined: Jul 2011 From: Texas Posts: 2,755 Thanks: 1405  Quote:
btw, a function $f(x)$ would also be 11 if $f'(x) < 0$ for all $x$ in its domain ... true?  
November 30th, 2016, 10:31 AM  #5  
Senior Member Joined: Nov 2015 From: United States of America Posts: 167 Thanks: 21 Math Focus: Calculus and Physics  Quote:
And funny that you mention that. I am in the math lab right now and that crossed my mind. I then came onto mmf and here you are pointing it out. But yes! That would make sense.  
December 13th, 2016, 08:55 PM  #6 
Senior Member Joined: Sep 2016 From: USA Posts: 383 Thanks: 207 Math Focus: Dynamical systems, analytic function theory, numerics 
Suppose $f(a) = f(b)$ and WLOG assume $a \leq b$. Then $a^3 + 2a = b^3 + 2b$ which can be rewritten as $a^3  b^3 + 2(ab) = 0$. Factoring $(ab)$ you get $(ab)(a^2 + ab+b^2 + 2) = 0$ and since $b^2 \geq ab$ it follows that $a^2 + ab + b^2 + 2 > 0$ implying $ab = 0$.


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