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Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions1 Mark
MCQ
Let $f : R - {n} \rightarrow R$ be a function defined by $\text{f(x)}=\frac{\text{x}-\text{m}}{\text{x}-\text{n}},$ where $\text{m}\neq\text{n.}$ Then,
  • A
    $f$ is one$-$one onto.
  • ✓
    $f$ is one$-$one into.
  • C
    $f$ is many one onto.
  • D
    $f$ is many one into.
✓

Answer

Correct option: B.
$f$ is one$-$one into.
Given function $f : R - {n} \rightarrow R$ be a function defined by $\text{f(x)}=\frac{\text{x}-\text{m}}{\text{x}-\text{n}},\ \text{m}\neq\text{n}$
If $f(x) = f(y)$ then
$\frac{\text{x}-\text{m}}{\text{x}-\text{n}}=\frac{\text{y}-\text{m}}{\text{y}-\text{n}}$
$\Rightarrow (x - m)(y - n) = (y - m)(x - n)$
After solving this we will get $x = y$
Hence, it is one$-$one.
$\text{f(x)}=\frac{\text{x}-\text{m}}{\text{x}-\text{n}},\ \text{m}\neq\text{n}$
$\text{y}=\frac{\text{x}-\text{m}}{\text{x}-\text{n}}$
$\Rightarrow y(x - n) = x - m$
$\Rightarrow yx - yn = x - m$
$\Rightarrow yx - x = ny - m$
$\Rightarrow x(y - 1) = ny - m$
$\Rightarrow\ \text{x}=\frac{\text{ny}-\text{m}}{\text{y}-1}$
Here, for $y = 1$ we can not define $x.$
Hence, it is not onto.

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