Q.Check the injectivity and surjectivity of the following function: $f:Z\to Z$ given by $f(x)=x^2$.

Two independent yes/no properties refine a plain function into something more specific. A function $f:A\to B$ is one-one (injective) if different inputs always give different outputs: $f(a)=f(b)\implies a=b$. Pictorially, no two arrows land on the same point of $B$; on a graph, this is exactly the horizontal line test — if no horizontal line meets the curve more than once, the function is one-one.

A function is onto (surjective) if every element of the co-domain $B$ is actually hit — has at least one pre-image in $A$ — equivalently, the range equals the whole co-domain. These two properties are independent: a function can be one-one but not onto (e.g. $f:N\to N,\ f(x)=x^2$ — never repeats a value, but misses every non-perfect-square), onto but not one-one, both at once (a bijection, which is exactly the condition needed for an inverse function to exist), or neither. …