The range of the function fx-x2-x-x2-2 x isA- R -1B- R -1-2- 1C- None of theseD- R

The correct option is B R { 1/2, 1} =x^2 x/x^2+2x Range of f Let f(x) = y, then f(x) = y ⇒x^2 x/x^2+2x=y ⇒ x^2 x = yx^2+2x ⇒ x^2 yx^2=2xy + x x^2(1 y)=x(2 ...

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Standard XII

Mathematics

Expressing x in Terms of y, to Find the Range of a Function

The range of ...

Question

The range of the function $f (x) = \frac{x^{2} - x}{x^{2} + 2 x}$ is

$R$

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$R - {1}$

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$R - {- \frac{1}{2}, 1}$

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None of these

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Solution

The correct option is C
$R - {- \frac{1}{2}, 1}$

$= \frac{x^{2} - x}{x^{2} + 2 x}$
Range of f Let f(x) = y, then
f(x) = y
$\Rightarrow \frac{x^{2} - x}{x^{2} + 2 x} = y \Rightarrow x^{2} - x = y x^{2} + 2 x \Rightarrow x^{2} - y x^{2} = 2 x y + x x^{2} (1 - y) = x (2 y + 1) x = \frac{2 y + 1}{1 - y}$
Clearly x will take real values, if $\frac{2 y + 1}{1 - y} \geq 0$
$\frac{2 y + 1}{y - 1} \geq 0$
$2 y + 1 > 0$ or $y - 1 < 0$
$y > - \frac{1}{2}$ or $y < 1$
$- \frac{1}{2} < y < 1$
Hence, range (f) $= [- \frac{1}{2}, 1] (- \frac{1}{2}, 1)$

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The range of the function f(x)=x2−xx2+2x is

The range of the function $f (x) = \frac{x^{2} - x}{x^{2} + 2 x}$ is