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Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions1 Mark
MCQ
Let the function $f : R - \{-b\} \rightarrow R - {1}$ be defined by $\text{f(x)}=\frac{\text{x}+\text{a}}{\text{x}+\text{b}},\ \text{a}\neq\text{b}.$ Then,
  • A
    $f$ is one$-$one but not onto.
  • B
    $f$ is onto but not one$-$one.
  • ✓
    $f$ is both one$-$one and onto.
  • D
    None of these.
✓

Answer

Correct option: C.
$f$ is both one$-$one and onto.
Injectivity: Let $x$ and $y$ be two elements in the domain $R - \{-b\},$ such that
$f(x) = f(y) \Rightarrow x + ax + b = y + ay + b$
$\Rightarrow x + ay + b = x + by + a$
$\Rightarrow xy + bx + ay + ab = xy + ax + by + ab$
$\Rightarrow bx + ay = ax + by$
$\Rightarrow a - bx = a - by$
$\Rightarrow x = y$
So, $f$ is one$-$one.
Surjectivity: Let $y$ be an element in the co$-$domain of $f,$
i.e., $R - {1},$ such that $f(x) = y$
$\Rightarrow x + ax + b = y$
$\Rightarrow x + a$
$\Rightarrow x = -a$
So, $f$ is onto.

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