I'm trying to find the domain for the function: square root of (9-x^2)...So what I do is solve for 9-x^2 >= 0 (b/c you can't have a negative number under a radical)...and get x is less than or equal to +3 and -3....
but the domain is supposedly [-3,3]...which would mean x > or = -3....but when I solved for it I got that x < or = -3...IT DOESN'T MAKE SENSE...
someone please explain
This is your error. You cannot "solve" an inequality like this just by taking the square root of both sides. Specifically, does give since you can add to both sides with changing the inequality sign. But you cannot then say " ".
When you are squaring, or taking the square root of both sides of an inequality, that is equivalent to multipying or dividing by an unknown- specifically you do not know whether that unknown is positive or negative and remember that multiplying or dividing both sides of an inequality by a negative number, the inequality sign reverses.
Once you have , I would recommend first solving the associated equation. That is, which does, in fact, have x= 3 and x= -3 as solutions. That divides the number line into 3 intervals, , , and on each interval of which the inequality is either true for every point in the interval or false for every point in the interval. (That is true because for any continuous function, f(x), f(x)< 0 can only change to f(x)> 0, and vice-versa, where f(x)= 0.)
Knowing that the three intervals are , , and . We need only look at a single point in each interval. x= -4 is in and so for all x in . x= 0 is in and so for all x in . Finally, is in (3, \infty) and so for all x in . Add to this the fact that and we have that if and only if x is in [-3, 3] or .
Going back to , another and perhaps simpler way to see that is this: if and only if and if and only if a< b. If x< -3 then it is also less than 3 so both of x- 3 and x+ 3= x-(3) are negative. Their product (the product of two negative numbers) is positive and so -(x- 3)(x+ 3) is [b]negative[/tex]. If -3< x< 3, then x- 3 is still negative but x+ 3= x-(-3) is now positive. Their product (the product of a positive and negative number) is negative so -(x- 3)(x+ 3) is positive. Finally, if x> 3, it is also greater than -3 so both x- 3 and x+ 3 are positive. The product of two positive numbers is positive so -(x- 3)(x+ 3) is negative. Again, we have that if and only if .
but the domain is supposedly [-3,3]...which would mean x > or = -3....but when I solved for it I got that x < or = -3...IT DOESN'T MAKE SENSE...
someone please explain