Analyzing a function's domain Given the function $$f(x)=\frac{1}{x-8}$$ and the question: What is the domain of $f(x)$ I would normally look for values which $f(x)$ can not take.
So I would check for plus infinity...Well $f(x)$ is defined positively for plus infinity...It is defined negatively for negative infinity and it is defined for 0 too...So I'm having a hard time finding a $f(x)$ which it can not take...I forgot how to do these, can anyone tell me how I analyze the domain of $f(x)$?
 A: We need to consider any/all $x$-values at which $f(x)$ is not defined.
What happens if $x = 8$?

Note: After re-reading your post and your comments below, I think you are confusing the "range" (image) of the function with it's domain. 
Loosely speaking:
The domain of a function is the set of all values that $x$ can take on, and consists of all and only those $x$ for which $f(x)$ is defined. 
The range of a function deals with the resultant "outputs" $f(x)$ when evaluated at each $x$ in the function's domain. This is the set $\{f(x)\mid x \in \text{ Domain }\}$
A: The domain is $(-\infty, 8) \cup (8, \infty)$, or equivalently, $\mathbb{R} \backslash \{8\}$. Note that the function is well-defined for all real numbers besides $8$. Often the first tactic one should take in determining the domain for these types of functions (fractional) is to set the denominator equal to zero and solve for its roots. Here, we have $$x - 8 = 0 \implies x=8$$
is a root of the denominator of $f(x)$, and thus a discontinuity of the function (so not included in its domain).
