Expert: Sherry Wallin - 1/8/2009

Question
sherry how can i finf the largest domail and range of a function.please tell me.


if  f(x)=√(16-x^2)  
then how can i finfd the largest domain and range of the function
please tell me


Answer
pratap, is it the sqrt(16) -x^2 or sqrt(16-x^2)?
When you talk about the domain and range there really isn't a largest. The domain is a set of values that x can be and the range is the set of values of y when you use the values of x. The best thing for you to do is to graph the equation and see what the domain and range is. I will assume you want
y = sqrt(16-x^2). The first thing you need to consider is where x is defined. The sqrt function is not defined for negative numbers so you need to find where sqrt(16-x^2)>=0. So in essence you are looking at 16 - x^2 >= 0 or 16 = x^2 which tells us that x is 4 or -4. This breaks up the domain into pieces. One piece is from negative infinity to -4, another from -4 to 4, and the last 4 to infinity. Choose a number in each of the pieces and see if it is indeed >= 0.
So I choose -5 in the first piece.
When x = -5 we get 16 -(-5)^2 = 16 - 25 = -9. Since the sqrt function cannot be negative the domain  is not between negative infinity and -4. Now try a number in the interval between -4 and 4, 0 is an easy one to calculate: 16 -0 = 16 which is greater than or equal to 0 so [-4,4] is part of the domain. Finally check a number in [4, infinity), say 5: 16-5^2 = -9 so x in 4 to infinity is not the domain of the function (because again the sqrt function cannot be negative because there is no numbers such that when you square it it is negative. All squared numbers are positive). Now that you know the only values for which x is >= 0 you can determine the range. So let f(x) = -4, i.e., 16 - (-4)^2 = 16- 16 = 0 and 16 -(4^2) is also 16-16 = 0. If you choose any other value in [-4,4] you will see that when x = 0 the largest value f(x) takes on is 16. So the domain is [-4,4] and the range is [0.16]. This is why I say the best way to do this is to graph the function and see what the range is.

Math Prof