Question

find the domain of
$\sqrt{x { }^{2} + 5x - 6}$
​

Answer 1

Answer:

(- ∞, - 6] U [1, + ∞)

Step-by-step explanation:

For the given function to exist, value under the square root must not be lesser than 0(or we say 'must not be -ve'). It can be 0 or greater than 0.

Case 1: f(x) ≥ 0

=> x^2 + 5x - 6 ≥ 0

=> x^2 + 5x - 6 ≥ 0

=> x^2 + 6x - x - 6 ≥ 0

=> x(x + 6) - (x + 6) ≥ 0

=> (x + 6)(x - 1) ≥ 0

=> x must lie in (- ∞, - 6] U [1, + ∞)

Method 2 :

Case 2 : when f(x) < 0

We can simply find the condition when value is lesser than 0 and *remove this from all possibilities. When f(x) < 0

=> x^2 + 5x - 6 < 0

=> x^2 + 6x - x - 6 < 0

=> x(x + 6) - (x + 6) < 0

=> (x + 6)(x - 1) < 0

=> x can be (- 6, 1).

Now remove this possibility from {R}.

=> x must be {R} - (-6, 1)

=> x must be (- ∞, + ∞) - (-6, 1)

=> x must be (- ∞, - 6] U [1, + ∞)