Question: Find the domain of the function. \[ f(x) = \frac{6x}{\sqrt{x + 2}} \] The domain is…

Find the domain of the function.

\[ f(x) = \frac{6x}{\sqrt{x + 2}} \]

The domain is \(\boxed{\quad}\).

(Type your answer in interval notation.)

Solution

To find the domain of the function \( f(x) = \frac{6x}{\sqrt{x} + 2} \), we need to ensure that the denominator is not zero and \( \sqrt{x} \) is defined. 1. Ensure \(\sqrt{x}\) is defined: \(\sqrt{x}\) is defined for \( x \geq 0 \). 2. Ensure the denominator is not zero: The denominator \(\sqrt{x} + 2\) must not be zero. \[ \sqrt{x} + 2 \neq 0 \] Solving for \(x\): \[ \sqrt{x} \neq -2 \] This inequality is always true since \(\sqrt{x}\) is never negative. Thus, the values of \(x\) that satisfy both conditions are \(x \geq 0\). The domain of the function in interval notation is \([0, \infty)\).