To find the domain of the function $f(x) = \sqrt{4 - x|x|}$, let's analyze it by considering two cases as suggested: when $x < 0$ and when $x > 0$.

Step 1: Understand the expression inside the square root

For the square root to be defined, the expression inside the square root must be non-negative. Therefore, we need:

$4 - x|x| \geq 0$

Case 1: $x > 0$

If $x > 0$, then $|x| = x$. Substituting this into the inequality:

$4 - x \cdot x \geq 0$ $4 - x^2 \geq 0$ $x^2 \leq 4$

Solving this inequality:

$-2 \leq x \leq 2$

Since we are considering $x > 0$ in this case, we narrow the solution to:

$0 < x \leq 2$

Case 2: $x < 0$

If $x < 0$, then $|x| = -x$. Substituting this into the inequality:

$4 - x(-x) \geq 0$ $4 + x^2 \geq 0$

This inequality is always true because $x^2 \geq 0$ for all real numbers $x$. Thus, for $x < 0$, the inequality is always satisfied, and $x$ can take any value less than 0.

Combining both cases

• For $x > 0$, we found that $0 < x \leq 2$.
• For $x < 0$, the inequality holds for all negative values of $x$.

Thus, the domain of $f(x)$ is:

$(-\infty, 2]$

Final Answer:

The domain of the function is $D = (-\infty, 2]$.

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Would you like more details on any part of the solution? Or do you have further questions?

Here are 5 related questions you could explore:

1. How does the behavior of absolute values affect the domain of a function?
2. What happens to the domain if the square root function is replaced with a different type of function, such as a logarithm?
3. How do you solve inequalities involving absolute values and square roots?
4. What is the difference between continuous and discontinuous functions in terms of their domains?
5. How can you graph functions with piecewise-defined domains?

Tip: Always check for restrictions caused by square roots, denominators, or logarithms when determining the domain of a function!