If the function is defined for all [tex] x \neq m [/tex], it means that [tex] m [/tex] is a point who doesn't belong to the domain of the function.
Now, since your function is a ratio, you can't have zero denominator. So, if a point nullifies the denominator, it can't belong to the function domain.
In your case, the denominator is [tex] x-2 [/tex], which is zero when
[tex] x-2=0\iff x=2 [/tex]
So, this function can't be evaluated at [tex] x=2 [/tex], and in other words, the function is defined when [tex] x \neq 2 [/tex], so that's the value for [tex] m [/tex].
