
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
November 30th, 2016, 08:50 AM  #1 
Member Joined: Nov 2015 From: Alabama Posts: 84 Thanks: 9 Math Focus: Trigonometry, Calculus, 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, 09:35 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,518 Thanks: 1240 
$f(x) = x^3 + 2x + 1$ $f'(x) = 3x^2 + 2 > 0 \text{ for all } x \in \mathbb{R} \implies f(x) \text{ is strictly increasing}$ ... how would this help to show $f(x)$ is 11? Look at the definition of an injective function. 
November 30th, 2016, 09:47 AM  #3  
Member Joined: Nov 2015 From: Alabama Posts: 84 Thanks: 9 Math Focus: Trigonometry, Calculus, Physics  Quote:
Appreciate it.  
November 30th, 2016, 09:55 AM  #4  
Math Team Joined: Jul 2011 From: Texas Posts: 2,518 Thanks: 1240  Quote:
btw, a function $f(x)$ would also be 11 if $f'(x) < 0$ for all $x$ in its domain ... true?  
November 30th, 2016, 11:31 AM  #5  
Member Joined: Nov 2015 From: Alabama Posts: 84 Thanks: 9 Math Focus: Trigonometry, Calculus, 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, 09:55 PM  #6 
Senior Member Joined: Sep 2016 From: USA Posts: 114 Thanks: 44 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 $ab = 0$.


Tags 
function, onetoone, show 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Show that sin(x^3) is not a periodic function.  suugakuedu  Algebra  2  December 27th, 2012 06:23 AM 
Show that the generating function  Solarmew  Applied Math  11  May 20th, 2012 07:34 PM 
show that the function f is bijection  zodiacbrave  Applied Math  1  February 23rd, 2012 09:02 PM 
Given function, show that  honeyman  Calculus  1  March 14th, 2010 09:31 AM 
show that f is identically the zero function  sailorsenshi  Complex Analysis  0  October 30th, 2007 12:10 AM 