What is the domain and range of #y = - sqrt(9-x^2)#?

1 Answer
Stefan V.
Aug 21, 2015

Domain: #[-3, 3]#
Range: #[-3, 0]#

Explanation:

In order to find the function's domain, you need to take into account the fact that, for real numbers, you can only take the square root of a positive number.

In other words, in oerder for the function to be defined, you need the expression that's under the square root to be positive.

#9 - x^2 >= 0#

#x^2 <= 9 implies |x| <= 3#

This means that you have

#x >= -3" "# and #" "x<=3#

For any value of #x# outside the interval #[-3, 3]#, the expression under the square root will be negative, which means that the function will be undefined. Therefore, the domain of the function will be #x in [-3, 3]#.

Now for the range. For any value of #x in [-3, 3]#, the function will be negative.

The maximum value the expression under the radical can take is for #x=0#

#9 - 0^2 = 9#

which means that the minimum value of the function will be

#y = -sqrt(9)= -3#

Therefore, the range of the function will be #[-3, 0]#.

graph{-sqrt(9-x^2) [-10, 10, -5, 5]}

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