I've been studying on Study Plan Practice, on MyMathLab for my College Algebra class. We're going over the Algebra of Functions right now and several things don't make much sense.
The question is: For the given functions, find the domain of $f$, $g$, $f+g$, and $(f+g)(x)$
$f(x) = x-5$, $g(x) = \sqrt{x-5}$
What is the domain of $f$?
I couldn't figure it out so I clicked Help me out. MyMathLab is telling me that $f = x-5 = f(x)$ The same thing with $g$, not $g(x)$. So are they the same? The other thing I don't get is what is different about $f+g$ and $(f+g)(x)$? Are they the same too? If $f = f(x)$ then it would logically follow that $f+g = (f+g)(x)$. Any pointers or is it really just a tautology?
A common convention is to impute a domain to an expression you would call its natural domain. Polynomial functions are defined everywhere, so their domains include all real numbers. I see a square root. You know the square root is defined when the thing under the radical is nonnegative. So, the domain of $g$ defined by $g(x) = \sqrt{x-5}$ is $[5,\infty).$ If you are adding two functions, both must be defined, so the domain of $f + g$ is the set of all points in the domain of both.