What is the domain and range of y=−√x2−3x−10?

1 Answer
Zor Shekhtman
Apr 27, 2016

Domain: the union of two intervals: x≤−2 and x≥5.
Range: (−∞,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: x2−3x−10≥0
Function y=x2−3x−10 is a quadratic polynomial with coefficient 1 at x2, it's negative between its roots x1=5 and x2=−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 +∞. So will the square root of it change. Therefore, taken with a negative sign, it will change from −∞ to 0.
Hence, the range of this function is (−∞,0].

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