TITLE: Is this expression even a function?
QUESTION [6 upvotes]: This is from Discrete Mathematics and its Applications

My question is on 22c. 
From the book, 
I inferred from the question that all of the listed mathematical expressions are functions.
Is the mathematical expression $(x + 1)/(x + 2)$ even a function though? What I got from my other question, 
Why is this not a function? was that 
a function needs to have exactly one output from every possible input in the domain. However this function doesn't have an output for $x = -2$, which is the domain of $\mathbb{R}$.

REPLY [7 votes]: You are essentially correct. While $f(x)=\frac{x+1}{x+2}$ is actually a function, its domain is not $\mathbb{R}$ since $f$ is not defined for $x=-2$. Thus it cannot be a function from $\mathbb{R}$ to $\mathbb{R}$, much less a bijection.

REPLY [4 votes]: You're correct that $f$ is not defined at $-2$.
It is still a function, though; its domain is $\mathbb R \setminus \{ -2 \}$, and its range is $\mathbb R \setminus \{ 1 \}$.
So we have, say,  $f : \mathbb R \setminus \{ -2 \} \to \mathbb R \setminus \{ 1 \}$.
Is this function a bijection?
Yes, it is; its inverse is
$$f^{-1}(y) = \frac{1 - 2y}{y - 1},$$
which is defined everywhere on the codomain of $f$. But is this function a bijection from $\mathbb R$ to $\mathbb R$, as the question asks?
Clearly not.