**Question:**

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ 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 (D) $f$ is neither one-one nor onto

**Solution:**

$f: \mathbf{R} \rightarrow \mathbf{R}$ is defined as $f(x)=x^{4}$.

Let $x, y \in \mathbf{R}$ 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,

$f(1)=f(-1)=1$

∴ *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.

∴ *f* is not onto.

Hence, function *f* is neither one-one nor onto.

The correct answer is D.

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