The graph of y- 1-x21-x2

Y= 1 x^21+x^2 y=0 ⇒ x= ± 1 x=0 ⇒ y=1 y( x)=y(x) ⇒ Graph is symmetric about Y axis y'= 4x(1+x^2)^2 y' gt;0 ⇒ x lt;0 nbsp; (increasing) y' lt;0 ⇒ x gt;0 ...

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Standard XII

Mathematics

Inequalities

The graph of ...

Question

The graph of $y = \frac{1 - x^{2}}{1 + x^{2}}$

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Solution

$y = \frac{1 - x^{2}}{1 + x^{2}}$
$y = 0 \Rightarrow x = \pm 1$
$x = 0 \Rightarrow y = 1$
$y (- x) = y (x)$
$\Rightarrow$ Graph is symmetric about $Y -$ axis
$y^{'} = - \frac{4 x}{(1 + x^{2})^{2}}$
$y^{'} > 0 \Rightarrow x < 0$ (increasing)
$y^{'} < 0 \Rightarrow x > 0$ (decreasing)
$y^{'} = 0 \Rightarrow x = 0$ (Local maxima)
$lim x \to - \infty y = lim x \to \infty y = - 1$

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The graph of y=1−x21+x2

y=1−x21+x2 y=0⇒x=±1 x=0⇒y=1 y(−x)=y(x) ⇒ Graph is symmetric about Y−axis y′=−4x(1+x2)2 y′>0⇒x<0 (increasing) y′<0⇒x>0 (decreasing) y′=0⇒x=0 (Local maxima) limx→−∞y=limx→∞y=−1

The graph of $y = \frac{1 - x^{2}}{1 + x^{2}}$