Question

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

Answer 1

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]`.