Question #5fa21

1 Answer
Jun 11, 2017

#{x in RR | x > 3 vee x < - 2}#

Explanation:

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}#.