January 17th, 2014, 07:18 PM  #1 
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Onetoone Functions
I'm supposed to answer whether or not the function is onetoone. 1. y = 6  ? 2. y = 4x  ? 3. f(x) = 4x + 4  ? 4. g(x) = ½ x + 2  ? 5. f(x) = x^3/5  When I take the derivative of this, I get 3/(5x^1/5)^2, which results in a positive number no matter what I plug in for x. So, it is 11. 6. g(x) = x^4/5 – 2  When I take the derivative of this, I get 4/(5x)^(1/5), which results in a negative number when plugging in a negative number for x and a positive number when plugging in a positive number for x. So, it is not 11 because there is a change in sign. When I take the derivative of problems 1, 2, 3, and 4 I'm left with a whole number. Where do I go from there? I feel like problems 5 and 6 are correct. 
January 17th, 2014, 08:36 PM  #2 
Member Joined: Feb 2013 Posts: 80 Thanks: 8  Re: Onetoone Functions
A function is injective (one to one) when each x value corresponds to a unique y value. That is, much like the vertical line test that they teach you in a high school functions class, you can use a horizontal line test to determine whether or not a function is injective. Also do note that one can make a function injective by limiting its domain (as frequently done with trigonometric functions). 
January 18th, 2014, 02:38 AM  #3  
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Re: Onetoone Functions Quote:
 
January 18th, 2014, 02:41 AM  #4  
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Re: Onetoone Functions Quote:
 
January 18th, 2014, 04:30 AM  #5 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Onetoone Functions
Are you saying that you are asked to determine whether or not functions are onetoone but don't know what "onetoone" means? There is no need to take derivatives. "Onetoone" means "if f(x)= f(y) then x= y" so, for example, if f(x)= 6 (a constant) then "f(x)= f(y)" becomes just "6= 6" and tells you nothing about x or y. No, f(x)= 6 is not "onetoone". If f(x)= 4x then "f(x)= f(y)" becomes 4x= 4y and, dividing both side by 4, x= y. Yes, f(x)= 4x is "onetoone". If (what you wrote would normally be interpreted as but it is clear from your work that that is not what you meant) the "f(x)= f(y)" becomes so taking the fifth power of both sides, and then taking the third root of both sides, x= y. Yes, is "onetoone". Now, had that been , there would have been a different answer. Taking the fifth power of both sides would now give and the square root of both sides is NOT "x= y". The square root of is x, not x. Even powers are NOT "onetoone" while odd powers are. That is because , for n and even number, is the same as while for n odd, it is not. 
January 18th, 2014, 09:39 AM  #6  
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Re: Onetoone Functions Quote:
 
January 18th, 2014, 11:05 AM  #7 
Member Joined: Dec 2013 Posts: 82 Thanks: 0  Re: Onetoone Functions
Would I be wrong in thinking that if then f is onetoone? Or do you need to be more explicit than this?

January 18th, 2014, 11:31 AM  #8  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,312 Thanks: 115  Re: Onetoone Functions Quote:
However, if f'(x) > 0 for all x, then f is strictly increasing, hence 11.  

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