find range of the function f defined by f(x)=(1/(1−x^2 ))


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Question Number 41765 by kunal1234523 last updated on 12/Aug/18

$f i n d r a n g e o f t h e f u n c t i o n f$ $d e f i n e d b y f (x) = \frac{1}{1 - x^{2}}$
	Answered by ajfour last updated on 12/Aug/18

	$R - [0, 1)$
	Answered by alex041103 last updated on 12/Aug/18

	$l e t s d e f i n e t h e i n v e r s e f u n c t i o n o f f (x)$ $a s F (f) . T h e n t h e d o m a i n o f F i s t h e$ $r a n g e o f f .$ $s o :$ $f = \frac{1}{1 - F^{2}}$ $\Rightarrow F^{2} = 1 - \frac{1}{f} \Rightarrow F (f) = \pm \sqrt{1 - \frac{1}{f}} .$ $T h e d o m a i n o f F i s d e f i n e d b y$ $1 - \frac{1}{f} ⩾ 0$ $\Rightarrow 1 ⩾ \frac{1}{f}$ $f o r f < 0 : 1 > \frac{1}{f} a s 1 > 0 .$ $f o r f > 0 : f ⩾ 1$ $\Rightarrow r a n g e i s f (x) \in (- \infty, 0) \cap [1, \infty)$
	Commented by kunal1234523 last updated on 13/Aug/18

	$s i r i s i t (- \infty, 0) \cup [1, \infty)$
	Commented by kunal1234523 last updated on 13/Aug/18

	$(- \infty, 0) \cap [1, \infty) = ϕ, i t h i n k$
	Commented by alex041103 last updated on 13/Aug/18

	$O h . . . Y e s . . .$
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