Question

If the function \(f:N\to N\) is defined as \(f(x)=x^2\), then \(f\) is:

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

Hint:

Squaring is injective on naturals but its range misses non-perfect-squares like 2.

Answer 1

The Correct Option is B

Step 1: Test injectivity with a concrete pair:
Suppose $f(3)=f(x)$ for some other natural $x$, i.e. $x^2=9$. The only natural solution is $x=3$ itself — no two distinct naturals square to the same value, so $f$ is one-one in general.

Step 2: Test surjectivity by listing the range:
The range of $f$ is $\{1,4,9,16,25,\dots\}$ — only perfect squares.

Step 3: Compare range to codomain:
The codomain is all of $N=\{1,2,3,4,\dots\}$, which includes non-squares such as $2,3,5,6,\dots$ that are never hit.

Final Answer:
Since the range is a strict subset of the codomain but distinct inputs never collide, $f$ is one-one but not onto. \[ \boxed{\text{One-one, not onto}} \]