I got this function, but I do not know how to approach the problem at all. Can someone guide me?
Determine the set of definitions of the function: $$f(x)=\frac{\sqrt{x^2-5x+6}}{|2x-1|}$$
Example
Let $f : [0,\infty) \to\Bbb{R}$ be a function by given formula $$f(x)=\frac{x}{(x+1)(x+2)}$$ The aswer would be $$[0, \infty) \subseteq \Bbb{R}\setminus\{−1, −2\}$$
In order to determine the domain of $f$, you need to determine all values where $f$ is not well-defined. Concretely, this means checking the values where you might accidentally divide by 0 or get negative values under a square root. Let's begin by assuming that the domain $D_0$ of $f$ is all of $\mathbb{R}$.
We see that $|2x-1|=0$ if and only if $x=\frac12$, so we exclude this number and now have that our domain is $D_1 := \mathbb{R} \setminus \{\frac12\}$. Next, you can check for yourself that $x^2-5x+6<0$ if and only if $2<x<3$. For these values of $x$, the term $\sqrt{x^2-5x+6}$ is not defined, so we will exclude these values from the domain as well. Hence, our domain is now $D_2 := \mathbb{R} \setminus \{ \{\frac12\} \cup (2,3) \}$. Since for all values in $D_2$ the function $f$ is well-defined, $D_2$ is the domain of $f$ that was to be determined. Hope this helps!
