# Is this function injective, surjective, or both?

1. Oct 3, 2009

### xpoferens

1. The problem statement, all variables and given/known data

The following function f is a function from R to R. Determine whether f is injective (one-to-one), surjective (onto), or both. Please give reasons.

2. Relevant equations

f(x) = (x+1)/(x+2) if x != -2
f(x) = 1 when x = 2

3. The attempt at a solution

f'(x) = 1/(x+2)2 > 0 for all x

and the limits at both infinities are 1 using l'hopital.

So the way I see it is the function grows from 1+ when x is a large negative, and then theres a horizontal assymptote at x = -2 so just before x = -2 f(x) tends to infinity and just after x = -2 the f(x) goes from negative infinity and gradually increases to 1. and then ofcourse at the point x = - 2 f(x) = 1 because that value has been forced in the definition.

So now by visualising the graph I have a strong suspicion that this function is bijective, but I have no idea how to prove it 'analytically'

2. Oct 3, 2009

### xpoferens

OK nevermind i figured it out. Is there a way to remove my original post? Or should I post my reasoning incase it could be helpful to others?

Last edited: Oct 3, 2009
3. Oct 3, 2009

### Office_Shredder

Staff Emeritus
Usually the latter is preferred