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: Testing monotonicity:
\(x^3\) is strictly increasing over all reals (and integers), since \(x_1<x_2 \Rightarrow x_1^3<x_2^3\); a strictly monotonic function is always injective.

Step 2: Testing the codomain coverage:
The range of \(f\) on \(Z\) is \(\{\ldots,-8,-1,0,1,8,27,\ldots\}\), a proper subset of \(Z\) (e.g. 5 is missed), so \(f\) does not cover the whole codomain \(Z\).

Final Answer:
Strictly increasing but not covering \(Z\): \(\boxed{\text{one-one, not onto}}\).