SOLUTION: let g(x)= log(subscript 2) (x^2+x-12). find the domain of g(X)

Question 382045: let g(x)= log(subscript 2) (x^2+x-12). find the domain of g(X)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

When finding domains of functions you want to restrict the possible values for x so that certain Mathematical "no-no's" cannot occur. Although there are more, the following are the three most common "no-no's" to avoid:

Zero denominators (or dividing by zero).
Negative radicands (expressions within a radical) of even-numbered roots (like square roots, 4th roots, etc.)
Negative (or zero) bases or arguments to logarithms.

Your function has no denominators nor does it have any even-numbered roots. But there is a logarithm so we must make sure that the base and the argument are positive. The base is 2 so there is no problem there. But the argument, , has a variable in it so we must restrict the domain so that the argument is always positive. The domain, therefore, is all the numbers that make the argument positive. IOW, the domain will be the solution to:

This is a quadratic inequality. Solving it start like solving a quadratic equation: Get one side equal to zero and factor the other side. We already have a zero on one side so we can go straight to factoring:
(x+4)(x-3) > 0
This inequality tells us that a product is positive. And how does a product of two numbers become positive? Answer: When the two numbers are both positive or when they are both negative. Saying this Mathematically:
(x+4 > 0 and x-3 > 0) or (x+4 < 0 and x-3 < 0)

We can simplify this a bit if we use some logic:

No matter what number x may be, x+4 will always be greater than x-3
If the smaller of two numbers is positive, then the larger one must also be positive.
If the larger of two numbers is negative, then the smaller one must also be negative.

If, after some thought, the above logic is not clear to you, then go ahead and solve the compound inequality above. But if it is clear to you then we can simplify/condense that inequality down to:
x+4 < 0 or x-3 > 0

No matter which compound inequality we solve we should end up with:
x < -4 or x > 3
Only these values for x will make the argument positive. So this is the domain of g(x). (Note: Domains may be specified like this or with set notation or with interval notation. You do not say in your post if there is a specific way to express the domain. I will leave it up to you to "convert" the above into the desired form.)
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