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RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS1 Mark
MCQ
Let $f: R \rightarrow R$ be defined as $f(x) = x^4$. Choose the correct answer.
  • A
    $f$ is one$-$one onto
  • B
    $f$ is many$-$one onto
  • C
    $f$ is one$-$one but not onto
  • ✓
    $f$ is neither one$-$one nor onto.
✓

Answer

Correct option: D.
$f$ is neither one$-$one nor onto.
$f: R \rightarrow R$ is defined as $f(x) = x^4.$
Let $\text{x},\text{y}\in\text{R}$ such that $f(x) = f(y).$
$\Rightarrow x^4 = y^4$
$\Rightarrow\text{x}=\pm\text{y}$
$\therefore f(x_1) = f(x_2)$ does not imply that $x_1 = x_2$
For instance,
$f(1) = f(-1) = 1$
$\therefore f$ is not one $-$ one.
Consider an element $2$ in co $-$ domain $R$.
It is clear that there does not exist any $x$ in domain $R$ such that $f(x) = 2.$
$\therefore f$ is not onto.
Hence, function $f$ is neither one$-$one nor onto.
The correct answer is $D$.

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