# How to establish a bijection

The question:

Show that the function $f : \mathbb{R} − \{−1\} → \mathbb{R} − \{2\}$ defined by $f(x) = \dfrac{4x + 3}{2x + 2}$ is a bijection, and find the inverse function.

How would I establish the function is a bijection, and how do I find the inverse using discrete math?

I understand a bijection means the function must both be onto and one-to-one, which it is, but how do you prove this?

As for the inverse, using algebra I have determined it to be $\dfrac{3-2x}{2x-4}$, but am I supposed to solve it some other way?

• Please, firstly establish your latex. – Jihad Oct 26 '14 at 3:46
• Edited, I think that's everything – user3362196 Oct 26 '14 at 3:53

## 1 Answer

The inverse function $g(x)$ should satisfy that $f\circ g = g\circ f =$ identity map. If such $g$ exists, then automatically $f$ is a bijection.

Now that you have already calculated the inverse, then check that the above condition is valid, and you are done.

Identity map is a function that sends $x$ to $x$ for every $x$ in its domain. In your case, $f\circ g$ means $x\mapsto g(x)\mapsto f(g(x))$ i.e. $x\mapsto\cfrac{3-2x}{2x-4}\mapsto f\left(\cfrac{3-2x}{2x-4}\right)=x$. This shows that $f\circ g$ is the identity map. Maybe try to prove $g\circ f$ by yourself?

• We were never taught about that map, can you link me to a site that explains it? Or if you are willing to explain it that would be great – user3362196 Oct 26 '14 at 4:03