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}#.
