find-range-of-the-function-f-defined-by-f-x-1-1-x-2-

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 \dots Y e s \dots