Question

Let $f:R\to R$ be defined as $f(x)=x^4$.
Choose the correct option.

• 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

Answer 1

Answer: (D)

Function $f:R\to R$ is defined as $f(x)=x^4$
Let $x,y\in R$ such that $f(x)=f(y)$
$\Rightarrow x^4=y^4$
$\Rightarrow x=\pm y$ (considering only real values)
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 codomain $R$. 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.