What is the domain and range of #y = -sqrt(x ^2 - 3x - 10)#?

1 Answer
Zor Shekhtman
Apr 27, 2016

Domain: the union of two intervals: #x<=-2# and #x>=5#.
Range: #(-oo, 0]#.

Explanation:

Domain is a set of argument values where the function is defined. In this case we deal with a square root as the only restrictive component of the function. So, the expression under the square root must be non-negative for the function to be defined.

Requirement: #x^2-3x-10 >= 0#
Function #y = x^2-3x-10# is a quadratic polynomial with coefficient #1# at #x^2#, it's negative between its roots #x_1=5# and #x_2=-2#.
Therefore, the domain of the original function is the union of two intervals: #x<=-2# and #x>=5#.

Inside each of these intervals the expression under a square root changes from #0# (inclusive) to #+oo#. So will the square root of it change. Therefore, taken with a negative sign, it will change from #-oo# to #0#.
Hence, the range of this function is #(-oo, 0]#.

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