Question

Let $f:R\to R$ be defined as $f\left(x\right)=x^4$. 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

Answer 1

Function $f:R\to R$ is defined as $f\left(x\right)=x^4$
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).