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

The Correct Option is D

\(f: R → R\) 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.