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

Apr 27, 2016

Domain: the union of two intervals: $x \le - 2$ and $x \ge 5$.
Range: $\left(- \infty , 0\right]$.

#### 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: ${x}^{2} - 3 x - 10 \ge 0$
Function $y = {x}^{2} - 3 x - 10$ 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: $x \le - 2$ and $x \ge 5$.

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