Is really $f(x)=\int g(x) dx$ a function? I saw many of this kind of questions on some text/question books. Is there any other explanation of this, or is it really wrong as I thought?
Here is a question of that kind: 
If $\displaystyle f(x)=\int x(x^2-a)^2 dx$ and $f(a)=7$ then $f(-a)=?$
Here what is $f(0)$ or $f(1)$? $\displaystyle f(0)=\int 0(0^2-a)^2 d0$ or $\displaystyle f(1)=\int 1(1^2-a)^2 d1$ does not make sense.
For me, a right function need to be as: $\displaystyle  f_c(x)=\int_c^xt(t^2-a)^2dt$
(where $c$ is some constant, can be $0$ as usual).
 A: You can consider
$$
\int x(x^2-a)^2\,dx
$$
as one of the functions $f(x)$ such that $f'(x)=x(x^2-a)^2$, which is nonetheless not determined until some further condition is imposed.
You do have a further condition, that is, $f(a)=7$, so the function is determined.
How do you determine it? You know that, for some constant $c$,
$$
f(x)=\frac{x^6}{6}-\frac{ax^4}{2}+\frac{a^2x^2}{2}+c
$$
and the condition $f(a)=7$ reads
$$
\frac{a^6}{6}-\frac{a^5}{2}+\frac{a^4}{2}+c=7
$$
thereby determining $c$.
However, this is not really necessary: the function $f$ satisfies the property that $f(x)=f(-x)$, so…
A: One usually uses the notation $\int f(x) \, dx$ to denote an anti-derivative, or, more rigorously, the collection of all possible anti-derivatives of the function $f(x)$ over some domain that it usually understood implicitly from the context. Thus, one can interpret the statement 
$$ \int x^2 \, dx = \frac{x^3}{3} + C $$
as the statement that all the possible anti-derivatives of the function $x^2$ on the real line are of the form $\frac{x^3}{3} + C$ for some $C \in \mathbb{R}$. This way, one can interpret the expression $f(x) = \int x(x^2 - a)^2 \, dx$ as the statement "$f(x)$ is one of the possible anti-derivatives of the function $x(x^2 - a)^2$ on $\mathbb{R}$" although admittedly, this can be written in a much less confusing way as "$f'(x) = x(x^2 - a)^2$ for all $x \in \mathbb{R}$".
The fundamental theorem of calculus states that if $f$ is continuous on an interval $(a,b)$, then an anti-derivative of $f$ is given by $\int_c^x f(t) \, dt$ where $c \in (a,b)$ is an arbitrary point but the notation $\int f(x) \, dx$ can be used without even knowing what is a definite integral, only knowing what is a derivative.
A: $f(x) = \int g(x) dx$ is very bad notation. The $x$ in the integral sign is a so-called 'dummy variable'. $\int g(x)dx$ is just a number, it does not depend on $x$. The only thing it  could possible mean is the function that has constant value $\int g(u)du$. When people erroneously write this down they usually mean 
$$f(x) = \int_a^x f(t)dt,$$
for some $a$. 
