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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS1 Mark
MCQ
The function $\text{f}:[0,\infty)\rightarrow\ \text{R}$ given by $\text{f(x)}=\frac{\text{x}}{\text{x}+1}$ is:
  • A
    One-one and onto.
  • ✓
    One-one but not onto.
  • C
    Onto but not one-one.
  • D
    Onto but not one-one.
✓

Answer

Correct option: B.
One-one but not onto.
Given function is $\text{f(x)}=\frac{\text{x}}{\text{x}+1}$ on $\text{f}:[0,\infty)\rightarrow\ \text{R}$
If f(x) = f(y)

$\Rightarrow\ \frac{\text{x}}{\text{x}+1}=\frac{\text{y}}{\text{y}+1}$

⇒ xy + x = xy + y

⇒ x = y

Hence, f is one-one.

If y = f(x)

$\text{y}=\frac{\text{x}}{\text{x}+1}$

⇒ xy + y = x

⇒ xy - x = -y

x(y - 1) = -y

$\text{x}=\frac{-\text{y}}{\text{y}-1}\neq\text{f(x)}$

It is not onto.

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