Questions: Find the domain of the function. (Enter your answer using interval notation.) f(x) = sqrt(x) + sqrt(5-x)

Solution Steps

To find the domain of the function \( f(x) = \sqrt{x} + \sqrt{5-x} \), we need to ensure that the expressions under both square roots are non-negative. This means \( x \geq 0 \) for the first square root and \( 5-x \geq 0 \) for the second square root. Solving these inequalities will give us the domain of the function.

The function \( f(x) = \sqrt{x} + \sqrt{5-x} \) consists of two square root functions: \( \sqrt{x} \) and \( \sqrt{5-x} \). The domain of a square root function \( \sqrt{u} \) is the set of all \( x \) such that \( u \geq 0 \).

1. Domain of \( \sqrt{x} \):

   \[ x \geq 0 \]

2. Domain of \( \sqrt{5-x} \):

   \[ 5-x \geq 0 \implies x \leq 5 \]

To find the domain of the entire function \( f(x) \), we need the intersection of the domains of \( \sqrt{x} \) and \( \sqrt{5-x} \).

• From \( \sqrt{x} \), we have \( x \geq 0 \).
• From \( \sqrt{5-x} \), we have \( x \leq 5 \).

The intersection of these two conditions is:

\[ 0 \leq x \leq 5 \]

The domain of \( f(x) \) in interval notation is the set of all \( x \) that satisfy both conditions:

\[ [0, 5] \]

Final Answer