Functions question - solving equation to find value of function

Question

Let $f : \mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f(0) = 1$ and for any $x$, $y\in\mathbb{R}$, we have:

$$f(xy+1) = f(x)f(y) - f(y) - x + 2$$

Find $f(x)$.

Answer 1

Just put $y=0$. You get $f(1)=f(x)-1-x+2$ for all $x$ so $f(x)=x+f(1)-1$. Then show that $f(1)$ has to be $2$.