Is this function injective, surjective, or both?

[xpoferens]

Homework Statement

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.

Homework Equations

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

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 there's 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'

 

[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?

 

[Office_Shredder]

Usually the latter is preferred