Question

Let $f: R→ R$ 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

Answer 1

Answer: (D)

f is neither one-one nor onto

Answer 2

$f: R → R$ is defined as $f(x)=x^4$.

Let $x, y ∈ R$ such that $f(x) = f(y)$.
$⇒ x^4=y^4$
$⇒ x=±y$
∴ $f(x_1)=f(x_2)$ 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.