Domain of a graph

Question

Do all values in the domain of a function need to be on the graph if the function is injective?

For example, is the function $f(x)=\dfrac{1}{(x-5)^2}$ injective if the domain is N plus? There is no value at $x=5$, so the function cannot be surjective, but does it satisfy the domain requirement?

Answer 1

Correction: $5$ cannot be in the domain of the function, since your function is not defined when the denominator is equal to zero.

The function is not injective. Note for example that $f(4) = f(6).$

Answer 2

You can't speak of surjectivity without explicity stating the range. That's the space in which the image lies, and it could be larger than the image. In fact, a function is surjective if and only if the image equals the range. So the rule can be the same, but if you enlarge the range, you will lose surjectivity.

Answer 3

We have $f(4)=f(6)=1$, so it's not injective.