find-range-for-1-f-x-x-2-9-x-1-2-f-x-ln-4-9x-2-

Question Number 115489 by ruwedkabeh last updated on 26/Sep/20

f i n d r a n g e f o r

1 . f (x) = \frac{\sqrt{x^{2} - 9}}{x - 1}

2 . f (x) = \ln (\sqrt{4 - 9 x^{2}})

Answered by PRITHWISH SEN 2 last updated on 26/Sep/20

1 . Let y = \frac{\sqrt{x^{2} - 9}}{x - 1}

x^{2} (y^{2} - 1) - {2 xy}^{2} + (y^{2} + 9) = 0

for real x

△ = {({2 y}^{2})}^{2} - 4 (y^{2} - 1) (y^{2} + 9) ⩾ 0

\Rightarrow (y - \frac{3}{2 \sqrt{2}}) (y + \frac{3}{2 \sqrt{2}}) ⩽ 0

Range f (x) \in [- \frac{3}{2 \sqrt{2}}, \frac{3}{2 \sqrt{2}}]

2 . Again let

y = \ln (\sqrt{4 - {9 x}^{2}})

\Rightarrow e^{2 y} = 4 - {9 x}^{2}

x = \pm \frac{1}{3} \sqrt{4 - e^{2 y}}

for x to be real

y ⩽ \ln 2

Range of f (x) \in (- \infty, \ln 2]

Commented by ruwedkabeh last updated on 26/Sep/20

T h a n k y o u v e r y m u c h!