Which interval is the domain of the function $g(x)=\sqrt{x-2}+2x$?

Key takeaway: For a function with a square root, the expression under the radical must be nonnegative.

We need the domain of

$g(x)=\sqrt{x-2}+2x$

Step 1: Check the square root

The radicand must satisfy

$x-2\ge 0$

so

$x\ge 2$

Step 2: Check the linear term

The term $2x$ is defined for all real $x$, so it adds no restriction.

Step 3: Write the domain

The only restriction is

$x\ge 2$

So the domain is

$\boxed{[2,\infty)}$

The correct choice is A.

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Pitfalls the pros know 👇 Do not treat $2x$ as part of the square root. Only the expression inside $\sqrt{x-2}$ determines the domain restriction. Another mistake is writing $(2,\infty)$; the endpoint 2 is included because $\sqrt{0}$ is defined.

What if the problem changes? If the function were $g(x)=\sqrt{x+2}+2x$, then the domain would be $x\ge -2$, or $[-2,\infty)$. If there were two square roots, you would need to satisfy both radicand conditions at the same time.

Tags: domain, radicand, square root function