Typo: in the definition of inverse image $E$ should be replaced by $H$
I'm reading Bartle and Sherbert: Introduction to Real Analysis.
The author introduces the definitions of direct and inverse images:
Let $f:A\rightarrow B$ be a function with domain $D(f)=A$ and range $R(f) \subseteq B$.
1.1.7 Definition If $E$ is a subset of $A$, then the direct image of $E$ under $f$ is the subset $f(E)$ of $B$ given by:
$$f(E):=\{f(x):x\in E\}$$
If $H$ is a subset of $B$, then the inverse image of $H$ under $f$ is the subset $f^{-1}(H)$ of $A$ given by:
$$f^{-1}(H):=\{x\in A:f(x)\in H\}$$
I also tried to understand using wikipedia but I had little success. The only pattern I could find similar to the one provided by wikipedia:
For example, for the function $f(x) = x^2$, the inverse image of $\{4\}$ would be $\{-2,2\}$.
Then I guess that the inverse image consists of all $x$'s that could result in the $x^2$, in this case. If the function was $f(x)=x$, the inverse image of $\{2\}$ would be $\{2\}$, right?
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$\begingroup$Inverse image preserves unions, intersections and complements. In topology and analysis we often work with families of subsets of the space, "open sets" or "Borel sets", and so on. These families are often closed under unions, or intersections - or some variation thereof.
Using the fact that inverse images preserves set operations means that we can use it to characterize "good" functions. Continuous functions are those that the preimage of an open set is open; Borel functions are those that the preimage of a Borel set is Borel; and so on.
For this reason we are usually more interested in the inverse image than in the direct image; however that too has its uses (but often coupled with some condition associated with inverse images). For example a bijection which is both open and continuous is in fact a homeomorphism.
$\endgroup$ $\begingroup$Your idea of "what is" the inverse image is correct. Given a function $f: A \to B$, the inverse image $f^{-1}(C)$ of some subset $C \subset B$ is the set of all points in $A$ which are mapped to the points in $C$ by the function $f$. You ask about the "purpose" of this. Well the point is: how you use this really depends on the context, this is just a convenient definition to speak about something that you'll need very often.
It's very common to speak about the subset of $A$ which is mapped to a certain subset of $B$, however it would get clumsy and tedious if every single time you would need to say: given the subset $C$ of $B$ let's consider the subset of $A$ which is mapped to $C$ by $f$. So, to make this simpler and to have an unified language you just say: given the inverse image $f^{-1}(C)$ and it'll be understood what you're saying.
Perhaps you want some examples on where this is used. Consider the circle of radius $r$ centered at the origin: $x^2+y^2=r^2$. Then if we define the function $f: \mathbb{R}^2 \to \mathbb{R}$ given by $f(x,y)=x^2+y^2-r^2$ then the circle is the set of points which are mapped to $0$ by $f$, in other words, the circle is $f^{-1}(\{0\})$. Also, when we talk about the inverse image of a set composed by just one element, we forget about set notation and just write $f^{-1}(0)$.
Also, the ellipsoid $(x/a)^2+(y/b)^2+(z/c)^2=1$ can be described using the function $f : \mathbb{R}^3 \to \mathbb{R}$ given by $f(x,y,z)=(x/a)^2+(y/b)^2+(z/c)^2-1$ so that it'll be the inverse image $f^{-1}(0)$.
A theorem then in differential geometry will grant that these objects are regular surfaces because that they can be described in this way. These are just two examples in geometry where you can put the inverse images to use, but inverse images are simply much more general and they're very convenient do describe many situations.
In summary: inverse images are just a way to talk about what is the set of elements in $A$ that are mapped to some subset of $B$. This is abstract and it'll only have one specific meaning when you use this within some specific study. The purpose then is just giving a name and notation to something common and useful.
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