document.write( "Question 244502: Find the domain of:\r
\n" ); document.write( "\n" ); document.write( "f(x)=sq root(x^2+x-12) \n" ); document.write( "
Algebra.Com's Answer #178845 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
When you are asked to find a domain of a function then
  1. Assume that all Real numbers are the domain.
  2. Then figure out what Real numbers must be excluded, if any. Numbers to be excluded are those, if any, which
    • Make a denominator zero
    • Make the radicand of any even-numbered roots (like square roots) negative
    • Make the argument to a known function invalid. For example, for the function f(x) = log(x+2), the domain is x > -2 because if x is less than or equal to -2, then the argument of the log is invalid. (All arguments to all log functions must be positive.)

\n" ); document.write( "The only one of these that your function has is an even-numbered root. So the domain is all Real numbers except those that make the radicand negative. In other words the domain is all Real numbers that make the radicand positive or zero. So we express this idea in the form of an inequality:
\n" ); document.write( "\"x%5E2%2Bx-12+%3E=+0\"
\n" ); document.write( "and solve it. The solution will be the domain of f(x).

\n" ); document.write( "Solving a quadratic inequality can be a little tricky (until you have done enough of them to get the idea). We start like we would if it was an equation:
\n" ); document.write( "Get one side equal to zero. (It already is.)
\n" ); document.write( "Factor the other side:
\n" ); document.write( "\"%28x%2B4%29%28x-3%29+%3E=+0\"
\n" ); document.write( "If this was an equation we would use the Zero Product Property to solve this. But this is not an equation.
\n" ); document.write( "Instead we have to think about how we can multiply two factors and end up with a positive result. (The \"or equal to zero\" part will be automatically handled during this solution so it will not be mentioned further.) I hope you understand that getting a positive result after multiplying two factors means that the factors are either both positive or both negative. So how do we express this? I hope you can see that the following compound inequality expresses this idea:
\n" ); document.write( "(\"x+%2B+4+%3E=+0\" and \"x+-+3+%3C=+0%29\" or ({{x + 4 <= 0}}} and \"x+-+3+%3C=+0\")

\n" ); document.write( "We could go ahead and solve the above. But instead I would like to introduce a simpler compound inequality and explain it:
\n" ); document.write( "\"x+-+3+%3E=+0\" or \"x+%2B+4+%3C=0\"
\n" ); document.write( "The reason these two inequalities can take the place of the four earlier inequalities is that we know which factor will always be the larger factor, no matter what x is. x+4 will always be larger than x-3! With this knowledge and a little logic we can reduce \"both factors are positive\" to \"x-3+%3E=+0\". This is so because if the smaller factor, x-3, is positive the larger one would have to be positive, too! And \"both factors are negative\" can be redeuced to \"x%2B4+%3C=0\" because if the larger factor, x+4, is negative then both the factors would have to be negative. I hope, after some thought, that this is clear because solving quadratic inequalities like this are much easier using the reduced compound inequalities.
\n" ); document.write( "So we will just solve
\n" ); document.write( "\"x+-+3+%3E=+0\" or \"x+%2B+4+%3C=0\"
\n" ); document.write( "giving:
\n" ); document.write( "\"x+%3E=+3\" or \"x+%3C=+-4\"
\n" ); document.write( "This is our domain.

\n" ); document.write( "P.S. It was the use of \"or equal to\" inequalities throughout that automatically handled the products that were zero. So even though we were talking mostly about positive and negative we used >=0 and <=0 instead of >0 and <0. \n" ); document.write( "
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