Find the domain of the function f(x) = sqrt(x - 4) + sqrt(x + 15). Give your answer in interval notation.

Step-by-step solution

1. 1

   Understand that a sum is defined only where every piece is defined. For $f(x)$ to produce a real number, both square roots must be real at the same $x$ — one of them being fine is not enough. So the domain is the intersection of the two individual domains, not their union. Since these are square roots (even index) and not cube roots, each radicand must be nonnegative; there is no denominator here, so unlike a root in a denominator, zero radicands are allowed.

2. 2

   Write the condition for the first radical.

   $x-4\geq 0\quad\Longrightarrow\quad x\geq 4$

3. 3

   Write the condition for the second radical.

   $x+15\geq 0\quad\Longrightarrow\quad x\geq -15$

4. 4

   Intersect the two conditions. Both must hold simultaneously, so keep the more restrictive one. Every $x\geq 4$ automatically satisfies $x\geq -15$, but not conversely — for example $x=0$ passes the second test and fails the first. Hence

   $x\geq 4$

5. 5

   Write the interval and check the endpoint. In interval notation the domain is

   $[4,\infty)$

   The bracket at $4$ is correct because $f(4)=\sqrt{0}+\sqrt{19}=\sqrt{19}$ is a perfectly good real number, whereas $f(3)$ would require $\sqrt{-1}$. So the domain includes its left endpoint and extends without bound to the right.