Find the values of x at which expression is defined

Question

$$f(x)=\frac{\left(x+1\right)\cdot \left(x-1\right)}{\left(x-1\right)\cdot \left(x-2\right)}$$

At which values of x, f(x) is defined?

I think at $x=1,2 $ is not defined. Is it correct or should we cancel $(x-1)$.

Answer 1

I have included the graph that was made by Symbolab free online calculator; you can clearly see your predictions to be true!

Answer 2

That's right, $x=1,2$ are not in the domain. Note that, your function $f:\mathbb{R}\setminus\{1,2\}\to\mathbb{R}$ can be written as $$f(x)=\frac{x+1}{x-2}$$ since we are excluded the value of $1$ and we can cancel out the both factors. This means, the graph of $f$ is the same as the graph of $g:\mathbb{R}\setminus\{2\}\to\mathbb{R}$, where $$g(x)=\frac{x+1}{x-2}$$ except that the point $(1,g(1))$ appears as a "hole" in the graph of $f$.