In mathematics exists something called 'abuse of notation', where you omit something when you know everyone reading it knows what you are meaning (or it is just impossible to have a crystal clear notation as in many diagramms in homological algebra), or we use the same notation for different objects. Again normally everyone knows what is meant, but it can be inconvenient for students new in the subject.
At the question with the function notation.
Basically you see it correctly.
There are different notations which are used.
For a function $f:X\to Y$ you can write $x\mapsto f(x)$, or $f(x)=y$ and this would mean exactly the same.
When first introduced to functions one would note it like $y=x$ or $y=x^2$. This notation has it flaws.
First of all the f(x)-notation can contain more information.
When we have $f(x)=x^2$ then $f(-1)=1$ and $f(1)=1$. So this notation gives us the point we are talking about (1|1) or (-1|1) while the $y=\dotso$ notation does not remember the point $x$, and only its value.
When first learning functions we create these value tables, where we keep track of $x$ and $y$.
The notation $y=\dotso$ should be viewed from a purley didactic point of view.
Making clear how to get an y-value and so on.
That these value tables are important, and how to scetch a graph.
As many problems at this level have to be solved by putting a given point in the equation.
Like calculate $m$ and $b$ in $y=mx+b$, when you have points given.
From the coordinate names it is clear where to put them.
Later when the student already is familiar with this, we can move on to a more fitting notation, which might be difficult for younger people.
When we care about other questions. What is the derivative, how to get extremum and so on.
However, students that learn the f(x) notation often do not really understand it.
They know what to do, but they would talk about the function f(x) not understanding that this is supposed to note a value on the y-axis.
Also do not treat this equation correctly.
One would see calculations like
$$f(x)=2x^2+4x+2 |:2$$
$$f(x)=x^2+2x+1 $$
when correct would be $\frac{f(x)}{2}=x^2+2x+1$, what comes from that most of the time the questions these equations have to answer is when $f(x)=0$, so it does not matter.
Just my two cents.
Best Answer
In the area of logic, $\longleftrightarrow$ is usually used for "if and only if" instead of $\iff$ (because who wants to bother drawing that second line all the time).
Otherwise when dealing with functions, $\longleftrightarrow$ might also be used to denote a bijective function. So $f \colon A \leftrightarrow B$ is a bijection between $A$ and $B$. Or you could similarly write $$ A \overset{f}{\longleftrightarrow} B $$
In regards to what was likely meant in the video that you saw, the following is true:
For a given value of $x$, one has $\sum\limits_{n=1}^\infty \frac{1}{n^x}$ converges if and only if $\int\limits_{1}^\infty \frac{1}{t^x}dt$ converges.