Question

The function \(f:Z\to Z\) defined by \(f(x)=x^3\ \forall x\in Z\) is:

• A. Onto
• B. Neither one-one nor onto
• C. One-one but not onto
• D. Many-one

Hint:

Test injectivity via strict monotonicity, then check whether every integer is a cube.

Answer 1

The Correct Option is C

Step 1: Checking one-one (injectivity):
If \(f(x_1)=f(x_2)\) then \(x_1^3=x_2^3\), and since the real cube root is unique, \(x_1=x_2\). So \(f\) is one-one.

Step 2: Checking onto (surjectivity):
For \(f\) to be onto \(Z\), every integer must be a perfect cube. But \(2\in Z\) has no integer \(x\) with \(x^3=2\). So \(f\) is not onto.

Final Answer:
\(f\) is \(\boxed{\text{one-one but not onto}}\).