For a square-root function, the domain is the set of inputs that give real outputs. Look at the expression inside the root: it cannot be negative. Test the boundary in Desmos, then solve an inequality to find every allowed input. A tempting mistake is to leave out the boundary, even though the square root of zero is defined.
Hints
- Hint 1
The radicand is the expression under the square root. Start where that expression equals . Which input cancels the , and does the function give a real output there?
- Hint 2
The domain includes every input that gives a real output, not only the boundary you tested. Require the whole expression under the root to be nonnegative. What inequality does that give you?
Step-by-step
Check the boundary, then solve the restriction
Step 1Check whether the boundary counts
The rule adds before taking the square root, so makes the expression under the root . Type , then in Desmos. It displays , so is an allowed input.
- Step 2
Write the condition for every input
The domain is every input that gives a real output. A square root is real when the expression under it is or positive, so the whole expression under the root must be nonnegative:
- Step 3
Find the allowed inputs
Subtract from both sides:
Inputs below make negative, while the boundary is included, as Desmos confirmed. The domain is all real numbers such that . Choice C.