I have no idea what this means and can't find anything after researching on the internet:-
$f: A\to B$ is a function under which each element of set $A$ has an image in set B
$f: x\mapsto y$ is a function under which $x$ is mapped to $y$.
Please help. Also, do the use of different words 'mapped' and 'image' indicate anything?
On
====short ====
"mapped to" is a verb. "image" is a noun. $A$ and $B$ are sets of elements. $x$ and $y$ are individual elements. Both sentences are saying the same thing. All the elements of $A$ are being mapped to some of the elements of $B$. When $x \in A$ is mapped to $y \in B$ we say that $y$ is the image of $x$ and the collection of all the "mapped to elements" is called "The image of $A$". We use the notation $f(A)$ to indicate all the mapped to points. Note: $f(A) \subset B$ but $B$ might (or might not) be bigger than $f(A)$.
==== long =====
$A$ is a collection of lots of points/numbers/elements. Each of these points/elements/numbers are mapped to a single point/element/number in the collection $B$.
The first sentence $f:A\to B$ says: we are mapping all the elements of $A$ to some or maybe to all elements of $B$.
The second sentence $f:x \mapsto y$ says: we are mapping the single element $x\in A$ to the single element $y \in B$. In other words, $f(x) = y$.
We can say that $x$ maps to $y$ or we can say that: when $x$ gets mapped, $x$ will have an image $y$ which is the element $x$ gets mapped to.
That is all about single individual elements.
Now $A$ is called the domain and every element of $A$ is mapped to some element of $B$.
$B$ is called either the range (if you learned math before 1985) or the co-domain (if you learned math after 1985), and it is the big set that contains all the elements that all of $A$ is mapped to.
The collection of all the points that $A$ gets mapped to is also called the image of $A$ and just as $A$ is the collection of all the points that are going to be mapped, the image of $A$ is the collection of all the images of points that have been mapped into $B$.
But note: Although all to elements of $A$ are mapped to $B$, not all the elements of $B$ need to be mapped to. $B$ might be bigger than than image of $A$ and have points that were not mapped to.
Example:
$f: \mathbb R \to \mathbb R$ and $f: x \mapsto x^2$. Every element of $\mathbb R$ gets mapped to an element of $\mathbb R$ but only non-negative elements are mapped to. There is no $x$ so that $x\mapsto -5$. So the image of $\mathbb R$ is $[0,-\infty)\subset\mathbb R$.
"$\mapsto$" describes what is happening on the individual element level. E.g., the notation $$f:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto x^2$$ describes the squaring function on the reals: the first bit tells us the domain and codomain, and the second bit tells us exactly what it does.
Note that both pieces are necessary (barring context clues): just writing "$f:\mathbb{R}\rightarrow\mathbb{R}$" wouldn't let me figure out that $f(2)=4$, and just writing "$f:x\mapsto x^2$" wouldn't tell me that $f$ is a function from reals to reals (as opposed, say, to being defined on the complex numbers).
As to "image" versus "mapped to," the distinction is grammatical: noun versus verb. We can say e.g. that the image of $2$ under $f$ (in the above example) is $4$, or that $2$ is mapped to $4$, but we can't say e.g. that the mapped to of $2$ is $4$ or that $2$ is imaged $4$.