Question 202423
The domain of a function is the set of all possible values for the input variable, which is usually "x". Generally, when a domain is not explcitly defined, the domain of a function is all Real numbers. However, one must exclude values which cause expressions which cannot be allowed.<br>
Examples of expressions which cannot be allowed:<ul><li>Division by zero</li><li>Negative radicands of even-numbered roots:<ul><li>{{{g(x) = sqrt(x + 3)}}} Since there are no square roots of negative numbers (within the set of Real numbers) the domain of g must ensure that (x+3) >= 0. In other "words", x >= -3.</li><li>{{{h(x) = root(6, 3x - 6)}}}. Since there are no 6th roots of negative numbers  (within the set of Real numbers) the domain of h must ensure that (3x - 6) >= 0. In other "words", x >= 2.</li><li>Note that {{{q(x) = root(3, 4x + 9)}}} has a domain of all real numbers since cube roots (in fact all odd-numbered roots) of negative numbers <b>do</b> exist within the set of Real numbers.</li></ul><li>Negative or zero agruments to logarithm functions (regardless of the base of the logarithm).</li><li>Arguments which are not allowed by certain other functions which are part of the definition of the function in question. An example of this would be f(x) = tan(x) + 4. Since the tan function is not defined for 90 degrees (or {{{pi/2}}} radians), these values must be excluded from the domain of f(x).</li></ul>
Now let's apply this to your problems.<br>
{{{m(x)=5/(x^2-9)}}}
Since we have a denominator we must avoid x-values that would make the denominator zero. So if we solve {{{x^2 - 9 = 0}}} we will find the x-values we must exclude from the domain. Factoring this equation we get {{{(x + 3)(x - 3) = 0}}}. From this we can see that the solution is x = 3 or x = -3.
So the domain of m(x) is all real numbers <b>except</b> 3 and -3.<br>

l(x)=5x-4
Since none of the items described above (denominators, even-numbered roots, logarithms, etc.) are present, there is nothing to exclude. The domain is all Real numbers.<br>

g(x)=7x=4/x=4
With 3 equal signs I'm not sure what this is. Since the "=" and the "+" are on the same key, I'm going to assume that the last two "=" are supposed to be "+".
If {{{g(x) = 7x + 4/x + 4}}} we must make sure the denominator of x does not become zero. So we must exclude 0 from the domain.
If instead {{{g(x) = 7x + 4/(x+4)}}} then we must make sure (x + 4) is not zero.  So x must not be -4.