what-is-range-of-function-f-x-x-x-2-1-

Question Number 83539 by jagoll last updated on 03/Mar/20

what is range of function

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

Commented by john santu last updated on 03/Mar/20

domain x^{2} - 1 > 0 \Rightarrow x < - 1 \lor x > 1

y \sqrt{x^{2} - 1} = x \Rightarrow (x^{2} - 1) y^{2} = x^{2}

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

Δ = {4 y}^{2} (y^{2} - 1) ⩾ 0

\Rightarrow y^{2} (y - 1) (y + 1) ⩾ 0

Range \Rightarrow y ⩽ - 1 \lor y ⩾ 1

Commented by john santu last updated on 03/Mar/20

Commented by mathmax by abdo last updated on 03/Mar/20

f (x) = \frac{x}{\sqrt{x^{2} - 1}} \Rightarrow f d e f i n e d o n] - \infty, - 1 [\cup] 1, + \infty [

{l i m}_{x \to - \infty} f (x) = {l i m}_{x \to - \infty} \frac{x}{∣ x ∣ \sqrt{1 - x^{- 2}}} = - 1

{l i m}_{x \to + \infty} f (x) = {l i m}_{x \to + \infty} \frac{x}{x \sqrt{1 - x^{- 2}}} = 1

f^{^{'}} (x) = \frac{\sqrt{x^{2} - 1} - x \times \frac{2 x}{2 \sqrt{x^{2} - 1}}}{x^{2} - 1} = \frac{2 (x^{2} - 1) - 2 x^{2}}{2 (x^{2} - 1) \sqrt{x^{2} - 1}} = \frac{- 2}{2 (x^{2} - 1) \sqrt{x^{2} - 1}}

= \frac{- 1}{(x^{2} - 1) \sqrt{x^{2} - 1}} < 0 \Rightarrow f i s d e c r e a s i n g o n D_{f}

x - \infty - 1 1 + \infty

f^{^{'}} - ∣∣ ∣∣ -

f - 1 d e c r - \infty + \infty d e c r 1

\Rightarrow f (] - \infty, - 1 [) =] - \infty, - 1 [a n d f (] 1, + \infty [) = [1, + \infty [