The domain of the function, f ( x )=( x +0.5) log (0.5+ x )(( x 2+2 x -3/4 x 2-4 x -3)) is

Question

The domain of the function, f ( x )=( x +0.5) log (0.5+ x )(( x 2+2 x -3/4 x 2-4 x -3)) is (A) ((-1/2), ∞) (B) [1,3] (C) ((1/2), 1) ∪((3/2), ∞) (D) ((-1/2), (1/2)) ∪((1/2), 1) ∪((3/2), ∞)

Tardigrade Admin · Accepted Answer

x+0.5>0 ⇒ x+0.5 ≠ 1 Also, (x2+2 x-3/4 x2-4 x-3)>0 ⇒ ((x+3)(x-1)/(2 x+1)(2 x-3))>0

Q. The domain of the function, $f (x) = (x + 0.5)^{l o g_{(0.5 + x)} (\frac{x ^{2} + 2 x - 3}{4 x ^{2} - 4 x - 3})}$ is

$(\frac{- 1}{2}, \infty)$

$[1, 3]$

$(\frac{1}{2}, 1) \cup (\frac{3}{2}, \infty)$

$(\frac{- 1}{2}, \frac{1}{2}) \cup (\frac{1}{2}, 1) \cup (\frac{3}{2}, \infty)$

$x + 0.5 > 0 \Rightarrow x + 0.5 \neq = 1$
Also, $\frac{x ^{2} + 2 x - 3}{4 x ^{2} - 4 x - 3} > 0$
$\Rightarrow \frac{( x + 3 ) ( x - 1 )}{( 2 x + 1 ) ( 2 x - 3 )} > 0$

Q. The domain of the function, f(x)=(x+0.5)log(0.5+x)​(4x2−4x−3x2+2x−3​) is

(2−1​,∞)

[1,3]

(21​,1)∪(23​,∞)

(2−1​,21​)∪(21​,1)∪(23​,∞)