Domain of the explicit form of the function y represented implicitly by the equation (1+x)cosy−x^2 =0 is (a) (−1,1] (b) (−1, 1−(√)5/2] (c) [1−(√)5/2, 1+(√)5/2] (d) [0, 1+(√)5/2]


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Question Number 39464 by nishant last updated on 06/Jul/18

$D o m a i n o f t h e e x p l i c i t f o r m o f$ $t h e f u n c t i o n y r e p r e s e n t e d$ $i m p l i c i t l y b y t h e e q u a t i o n$ $(1 + x) c o s y - x^{2} = 0 i s$ $(a) (- 1, 1] (b) (- 1, 1 - \sqrt{} 5 / 2]$ $(c) [1 - \sqrt{} 5 / 2, 1 + \sqrt{} 5 / 2]$ $(d) [0, 1 + \sqrt{} 5 / 2]$
	Answered by MJS last updated on 06/Jul/18

	$\cos y = \frac{x^{2}}{x + 1} \Rightarrow x \neq - 1 \land \frac{x^{2}}{∣ x + 1 ∣} ⩽ 1$ $x^{2} ⩽∣ x + 1 ∣$ $f (x) = x^{2} - x - 1 = 0 \Rightarrow x_{1} = \frac{1}{2} - \frac{\sqrt{5}}{2}; x_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2}$ $f (x) = x^{2} + x + 1 = 0 \Rightarrow no real solution$ $(\frac{1}{2} - \frac{\sqrt{5}}{2} ⩽ x ⩽ \frac{1}{2} + \frac{\sqrt{5}}{2})$ $domain is [\frac{1}{2} - \frac{\sqrt{5}}{2}; \frac{1}{2} + \frac{\sqrt{5}}{2}]$
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