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In Bert Mendelson's "Introduction to Topology" image is defined as follows:

Let $f: A \to B$

For each subset $X \subset A$, the subset of B whose elements are the points $f(x)$ such that $x \in X$ is denoted by $f(X)$. $f(X)$ is called the image of X.

It means that there can be many images. Consider function $f(x) = x$. It's domain and range is R. Lets take X={1,2}. Images of X are sets {1}, {2}, {1,2} - definition holds true for them, so every one of them is f(X). Is this correct? Clearly I'm missing something here.

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    $\begingroup$ No, the image of $X$ (under $f$) is the set $\left\{ b \in B : \bigl(\exists x \in X\bigr)\bigl(f(x) = b\bigr)\right\}$. The subset whose elements are the points ... It would be clearer with a formula given in addition to the verbal description. $\endgroup$ – Daniel Fischer Nov 2 '13 at 16:21
  • $\begingroup$ Thanks, I get it now. Will post answer with reference to your comment in case there will be no answer. $\endgroup$ – pusheax Nov 2 '13 at 16:56
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The plays a crucial role in this definition as pointed out by Daniel Fisher in his comment:

No, the image of X (under f) is the set {b∈B:(∃x∈X)(f(x)=b)}. The subset whose elements are the points ... It would be clearer with a formula given in addition to the verbal description.

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