We have: f(x) = frac(1)(sqrt((x)^(2) - (x) - 6))
The argument of a square root must be greater than or equal to zero.
Also, the denominator of a fraction cannot be equal to zero.
Let's use these conditions to find the largest possible domain of f(x):
Rightarrow sqrt(x^(2) - x - 6) > 0
Squaring both sides of the equation:
Rightarrow (sqrt(x^(2) - x - 6))^(2) > 0^(2)
Rightarrow x^(2) - x - 6 > 0
Then, let's factorise the quadratic equation using the "middle-term break":
Rightarrow x^(2) + 2 x - 3 x - 6 > 0
Rightarrow x (x + 2) - 3 (x + 2) > 0
Rightarrow (x + 2)(x - 3) > 0
Rightarrow x + 2 > 0 and x - 3 > 0
Rightarrow x > - 2 and x > 3
or
Rightarrow x + 2 < 0 and x - 3 < 0
Rightarrow x < - 2 and x < 3
therefore x > 3 or x < - 2
Therefore, the largest possible domain of f(x) is {x in RR | x > 3 vee x < - 2}.