What does it mean for a function to be decidable? (homework) Note: This is part of a homework exercise. I am asking for clarifications, not a solution!

Given: Assume $g: \mathbb{N} \mapsto \mathbb{N}, g\in R$ ($R$ is the class of recursive functions) is a strictly monotone function, i.e., $\forall n\in \mathbb{N}: g(n)  &lt; g(n+1)$. 

Prove that $g(\mathbb{N})$ is decidable. 

Clarification of the assignment:
I am not sure I understand the exercise correctly. What does it mean that a function is decidable? I assume that it's a decision problem of the form "For $x \in \mathbb{N}$, is $x\in range(g)$? For a function $g$ to be decidable, we can then define a decision procedure that returns $yes$ or $no$ for any $x$. 

Is this correct? Or am I on the wrong track?
They are not asking to prove that the function $g$ is decidable, but the image $g(\mathbb{N})$ is decidable. The image is a subset of $\mathbb{N}$, namely
$$g(\mathbb{N}) = \{ k \in \mathbb{N} \mid \exists n \in \mathbb{N} . g(n) = k \}.$$
The question reduces to showing decidability of "given $k$, does there exist $n \in \mathbb{N}$ such that $g(n) = k$?".