`f(x)=sqrt(2-abs(x))+sqrt(1+abs(x))`. Find the domain of f(x).

Answer

To find the domain of the function \( f(x) = \sqrt{2 - |x|} + \sqrt{1 + |x|} \), we need to ensure that the expressions under both square roots are non-negative.

### Step 1: Analyze the first square root

The first part of the function is \( \sqrt{2 - |x|} \). For this square root to be defined, the expression inside must be non-negative:

\[ 2 - |x| \geq 0 \]

This simplifies to:

\[ |x| \leq 2 \]

### Step 2: Solve the inequality

The inequality \( |x| \leq 2 \) means that \( x \) must lie within the interval:

\[ -2 \leq x \leq 2 \]

### Step 3: Analyze the second square root

The second part of the function is \( \sqrt{1 + |x|} \). For this square root to be defined, the expression inside must also be non-negative:

\[ 1 + |x| \geq 0 \]

### Step 4: Solve the inequality

Since \( |x| \) is always non-negative, \( 1 + |x| \) is always greater than or equal to 1. Thus, this part does not impose any additional restrictions on \( x \).

### Step 5: Determine the overall domain

The domain of \( f(x) \) is determined by the intersection of the domains from both parts. The first part restricts \( x \) to the interval \( [-2, 2] \), while the second part allows all real numbers. Therefore, the domain of the function \( f(x) \) is:

\[ \text{Domain of } f(x) = [-2, 2] \]

### Final Answer

Thus, the domain of the function \( f(x) = \sqrt{2 - |x|} + \sqrt{1 + |x|} \) is:

\[ [-2, 2] \]

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