Find domain of the following functions f x = x + 0 . 5 l o g 0 . 5 + x x 2 + 2 x - 3 4 x 2 - 4 x - 3 .

Given, f x = x + 0 . 5 l o g 0 . 5 + x x 2 + 2 x - 3 4 x 2 - 4 x - 3 log is defined only when, x 2 + 2 x - 3 4 x 2 - 4 x - 3 > 0 & 0 . 5 + x > 0 & 0 . 5 + x ≠ 1 ⇒ x 2 + 3 x - x - 3 4 x 2 + 2 x - 6 x - 3 > 0 & x > - 1 2 & x ≠ 1 2 ⇒ x - 1 x + 3 2 x - 3 2 x + 1 > 0 & x > - 1 2 & x ≠ 1 2 For, x - 1 x + 3 2 x - 3 2 x + 1 > 0 , ⇒ x ∈ - ∞ , - 3 ∪ - 1 2 , 1 ∪ 3 2 , ∞ Taking intersection of all above cases we get, x ∈ - 1 2 , 1 2 ∪ 1 2 , 1 ∪ 3 2 , ∞ We know that exponential functions are defined for all real values x ∈ - ∞ , ∞ . Hence, the domain of f x is x ∈ - 1 2 , 1 2 ∪ 1 2 , 1 ∪ 3 2 , ∞ .

If a function is defined as f x = log h x g x , where g x = sinx + sinx , h x = sinx + cosx , 0 ≤ x ≤ π . Then find the domain of f x .

Given, f x = log h x g x , where g x = sinx + sinx , h x = sinx + cosx , 0 ≤ x ≤ π We know that in t , t ≥ 0 Here, log h x g x ≥ 0 We know that log a b ≥ m ⇒ b ≥ a m , a > 1 b ≤ a m , 0 < a < 1 Case-I : When 0 < sin x + cos x < 1 ⇒ 0 2 < 1 2 sin x + 1 2 cos x < 1 2 ⇒ 0 < sin x . cos π 4 + cos x . sin π 4 < 1 2 ⇒ 0 < sin x + π 4 < 1 2 ⇒ 0 < x + π 4 < π 4 ⇒ x ∈ - π 4 , 0 But it is given that 0 ≤ x ≤ π So, this case is rejected. Case-II : When sin x + cos x > 1 ⇒ 1 2 sin x + 1 2 cos x > 1 2 ⇒ cos π 4 . sin x + sin π 4 . cos x > 1 2 ⇒ sin x + π 4 > 1 2 ⇒ π 4 < x + π 4 < 5 π 4 ⇒ 0 < x < π 2 It is given 0 ≤ x ≤ π . Hence, 0 < x < π 2 ⇒ g x ≥ h x 0 for 0 < x < π 2 ⇒ g x ≥ 1 ⇒ sin x + sin x ≥ 1 for 0 < x < π 2 We know that for 0 < x < π , sin x > 0 ⇒ sin x = + sin x ⇒ 2 sin x ≥ 1 for 0 < x < π 2 ⇒ sin x ≥ 1 2 ⇒ x ∈ π 6 , 5 π 6 but 0 < x < π 2 ⇒ x ∈ π 6 , π 2 Now, log is defined only when g x > 0 ⇒ sin x + sin x > 0 For 0 ≤ x ≤ π , sinx = + sinx ⇒ 2 sin x > 0 ⇒ x ∈ 0 , π And h x > 0 and h x is not equal to 1 ⇒ sin x + cos x > 0 and sin x + cos x is not equal to 1 ⇒ 2 1 2 sin x + 1 2 cos x > 0 and 2 1 2 sin x + 1 2 cos x is not equal to 1 ⇒ sin x + π 4 > 0 and 2 sin x + π 4 is not equal to 1 ⇒ x ∈ 0 , 3 π 4 and x is not equal to 0 or π 2 Now taking intersection of all cases, we get x ∈ π 6 , π 2 Hence, the domain of f x for 0 ≤ x ≤ π is π 6 , π 2 .