The range of the function
Question:

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

(a) R

(b) R − {1}

(c) R − {−1/2, 1}

(d) None of these

Solution:

(c) R − {-1/2,1}

$f(x)=\frac{x^{2}-x}{x^{2}+2 x}$

Let $y=\frac{x^{2}-x}{x^{2}+2 x} \quad[$ Also,$x \neq 0]$

$\Rightarrow y=\frac{x(x-1)}{x(x+2)}$

 

$\Rightarrow y=\frac{(x-1)}{(x+2)}$

$\Rightarrow x y+2 y=x-1$

$\Rightarrow x=\frac{2 y+1}{1-y}$

Here, $1-y \neq 0$

or, $y \neq 1$

Also, $x \neq 0$

$\Rightarrow \frac{2 y+1}{1-y} \neq 0$

 

$\Rightarrow y \neq-\frac{1}{2}$

Thus, range $(f)=\mathrm{R}-\left\{-\frac{1}{2}, 1\right\}$.

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