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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?

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1 Answer 1

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

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  • $\begingroup$ Thanks thats what i was also thinking $\endgroup$ May 16 at 19:35
  • $\begingroup$ @ProblemDestroyer No problem. $\endgroup$
    – MathGeek
    May 17 at 13:33

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