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Relations and Functions — Maths STD 12 Science — Question

Gujarat 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 $x, y \in R$ such that $f(x)=f(y)$.
$\Rightarrow x^4=y^4$
$\Rightarrow x= \pm y$
$\therefore f \left( x _1\right)= f \left( x _2\right)$ 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 $D$.
d. $f$ is neither one-one nor onto.

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