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RELATIONS AND FUNCTIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRELATIONS AND FUNCTIONS1 Mark
MCQ
The range of the function $\text{f(x)}=\frac{\text{x}^2-\text{x}}{\text{x}^2+2\text{x}}$ is:
  • A
    $\text{R}$
  • B
    $\text{R}-\{1\}$
  • ✓
    $\text{R}-\Big\{\frac{1}{2},1\Big\}$
  • D
    None of these.
✓

Answer

Correct option: C.
$\text{R}-\Big\{\frac{1}{2},1\Big\}$
$\text{f(x)}=\frac{\text{x}^2-\text{x}}{\text{x}^2+2\text{x}}$
Let, $\text{y}=\frac{\text{x}^2-\text{x}}{\text{x}^2+2\text{x}}$ $\big[\text{Also},\text{ x}\neq0\big]$
$\Rightarrow\text{y}=\frac{\text{x}(\text{x}-1)}{\text{x}(\text{x}+2)}$
$\Rightarrow\text{y}=\frac{(\text{x}-1)}{(\text{x}+2)}$
$\Rightarrow\text{xy}+2\text{y}=\text{x}-1$
$\Rightarrow\text{x}=\frac{2\text{y}+1}{1-\text{y}}$
Here, $1-\text{y}\neq0$
Or, $\text{y}\neq1$
Also, $\text{x}\neq0$
$\Rightarrow\frac{2\text{y}+1}{1-\text{y}}\neq0$
$\Rightarrow\text{y}\neq-\frac{1}{2}$
Thus, range $\text{(f)}=\text{R}-\Big\{-\frac{1}{2},1\Big\}$

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