What can be a function where $x \neq 2, y \neq 1$ for all $x,y$?

Question

I am trying to find a possible function to this graph below.

I am really bad with graphs so if anyone can further elaborate the ways to help identify graphs, I will deeply appreciate it.

Answer 1

Hint: Looks like the plot of $xy=1$, but translated.

Also, this looks like a homework so please it a try first.

Answer 2

You might notice that the function

$$y = \frac 1 x$$

looks "kind of" like this, except $x,y$ can never be zero. On top of that, you play with the expression a little, you might notice that, if we have

$$y = \frac{1}{x-a}$$

for any particular $a$, then $x$ can never be equal to $a$ because the denominator is undefined.

And then you realize for the $y=1/x$ equation that "$y$ cannot be this value $b$" is just translating the graph of $y$ by $b$ units vertically. And thus,

$$y = \frac{1}{x} + b$$

gives you a general function where $x$ is never $0$, $y$ is never $b$.

How might you combine these two ideas to form a function such that $x \neq a, y \neq b$ for all $x$ and $y$?