# Definition of image of an element; what does it mean to say that a function takes all values of subset?

Wikipedia Article on topic of images that

" Given $$y$$, the function $$f$$ is said to "take the value $$y$$ " or "take $$y$$ as a value" if there exists some $$x$$ in the function's domain such that $$f(x)=y$$. Similarly, given a set $$S, f$$ is said to "take a value in $$S^{\text {" }}$$ if there exists some $$x$$ in the function's domain such that $$f(x) \in S$$. However, " $$f$$ takes [all] values in $$S^{\text {" }}$$ and " $$f$$ is valued in $$S^{\prime \prime}$$ means that $$f(x) \in S$$ for every point $$x$$ in $$f^{\prime}$$ s domain."

My Query
It is related to the the meaning of $$"f$$ take all values in $$S"$$. Does this means $$S$$ is the range? Or $$S$$ is a subset of actual range of $$f?$$ In my opinion, former one is correct because it is being said that for all $$x$$ belongs to $$f$$, domain $$f(x)$$ is belonging to $$S$$ or is it that $$S\subseteq f(x)$$ range?

$$S$$ is the range of $$f(x)$$. For example, for $$sin(\theta$$) , $$S=\mathbb C$$ (yes, you can define $$sin(\theta)$$ for the complex numbers).