What does $f(x)$ denote? All
I have been reading from my textbook, and I am wondering if my textbook is just badly written since I have learned in the past that $f(x)$ means $y$ (output) and the variable $x$ is the input. However, the book is using $f(x)$ as the expression. 
For example, Evaluate if $f(x) = x^3$ is odd. 
Now, I place $-x$ within the parentheses and cube it. Thus, I get $-x^3$; however, I am told by the book that I can take the negative sign outside of the term $x^3$ and then it will become $-(x^3)$ and thus I can write it as $-f(x)$. 
How can this be?  I thought $f(x)$ stood for the $y$ value only? 
Please explain as if I am a 5 year old. 
 A: $f(x)$ is just a notation for the output of a function $f$ when given a value $x$. So I take $x$ and this function $f$ gives me a number $f(x)$. That's it. So when we define $f(x)$ as $x^3$ then when we input $-x$ into $f$ we get: $f(-x)=-x^3$. But that number $x^3$ is by definition our function $f(x)$. So we can write $f(-x)$ as $-f(x)$.
I hope this helps.
Best wishes, $\mathcal H$akim.
A: Maybe Im going to say the obvious, but here we go.
$$f(-x) = (-x)^3 = (-1\cdot x)^3 = (-1)^3\cdot x^3 = -1\cdot x^3 = -x^3$$
Therefore, $f(-x) = -x^3 = -f(x)$, this is just a property about $f(x)$.
A: $f$ is a variable that denotes a function.
$f(x)$ is an expression that denotes a value: specifically, the value that is the output of $f$ when plugging in $x$ as the input.
Unfortunately, it is awkward to define a function $f$ directly. More common is to give a pointwise definition of $f$: e.g. define
$$ f(x) = x^3$$
Knowing this equation (between the variable value $f(x)$ and the variable value $x^3$) holds for all possible values of $x$ is sufficient information to uniquely identify a specific function $f$.
Unfortunately, this is still awkward for many purposes. Many people will use the expression $x^3$ when they need to denote a function rather than a variable quantity. This means reading mathematics can be sometimes ambiguous.
The only standard notation to allow this style while remaining unambiguous is lambda calculus: if someone writes $\lambda x.x^3$, they are referring to the function that is the same as the function $f$ defined above. Interpret this expression as saying "the function that, on input $x$, gives $x^3$". But if they merely write $x^3$ by itself, they are referring to a value and not a function.
Alas, this is not in general use.
A: The expression $$f:\forall x\in\mathbb{R}, \exists f(x)|f(x)=x^3$$
is read "$f$ such that for all/for each/for every/for any x in  $\mathbb{R}$, there exists some $f(x)$ such that $f(x)=x^3$".

Such statements feature not only the condition ($f(x)=x^3$), but also the universal quantifier ($\forall x \in \mathbb{R}$) and an existential quantifier ($\exists f(x)$).

Unfortunately, in High School, teachers say that $f$ is the function and $f(x)$ is the value of the function $f$ evaluated at $x$. They then go on to say that the function is defined as $f(x)=x^3$.
Either $f(x)$ is the function or $f(x)$ is the value of the function $f$ evaluated at $x$. You can't have it both ways, Mr. Math Teacher. Stop confusing your students, and start using existential and universal quantifiers.
Don't be spooked by the symbols $\forall$ and $\exists$; they just stand for for all and there exists, respectively. And be aware that the expression
$$f:\forall x\in\mathbb{R}, \exists f(x)|f(x)=x^3$$
can be rewritten in the more englishy form
$$f:\:for\:every\:x\in\mathbb{R}, there\:exists\:some\:number\:f(x)\:|\:f(x)=x^3$$
keeping the : and | to mean such that
