Question 709749
Sometimes the domain of a function is given to you. But when you are asked to find the (implicit) domain then you:<ol><li>Start by assuming that the domain is all real numbers.</li><li>Look for x values that "cannot be". If you find some then exclude these values from the domain.</li></ol>What kinds of x values "cannot be"? What kinds of x values must be excluded from the domain? Answer: There are a variety of "no-no's" in Math:<ul><li>Zero denominators. Any x that would make a denominator in the function turn into zero must be excluded.</li><li>Negative radicands of even-numbered roots. ("Radicand" is the name for the expression inside a radical. "Even-numbered roots" means square (or 2nd) roots, 4th roots, 6th roots, etc.) Since you cannot get a negative result when raising a real number to an even power, we cannot have negative radicands in even-numbered roots. Any x values that would make the radicand of an even-numbered root turn negative must be excluded from the domain.</li><li>Invalid arguments or bases of logarithms. Valid arguments to logarithms are positive numbers. Valid bases for logarithms are positive but not a 1. And x value that makes the argument or base of a log be invalid must be excluded from the domain.</li><li>Certain arguments to certain Trig functions and inverse Trig functions are invalid/undefined. And x value that would make the argument of any Trig function (or inverse) invalid/undefined must be excluded from the domain.</li></ul>These are the most "popular" reasons for excluding x values from the domain. In a nutshell: <i>Any</i> x value(s) that make some part of a function's definition invalid/undefined must be excluded from the domain of the function.<br>
Your function has no denominators, even-numbered roots or Trig functions. But it does have logarithms. The bases have no x's in them so they will be valid no matter what x is. But the arguments have x's in them so we must make sure they are valid. We need all three arguments to be positive. So:
2x + 2 > 0 and x + 3 > 0 and x > 0
Solving these we get:
x > -1 and x > -3 and x > 0
These all say x is greater than something. In this situation we can replace all three with just one: x > 0. (Take a moment to see why this is so. If x > 0, won't it automatically be greater than -1 and -3? On the other hand, if x > -1 then x will be greater than -3 but it won't necessarily be greater than zero. (e.g. -1/2 is greater than -1 but not greater than zero.))<br>
So the domain of f(x) is all numbers greater than zero (IOW: all positive numbers).