https://file.aiquickdraw.com/mathbot/file/1726062611425janzq4u7.jpg

The function shown in the image is:

\[
f(x) = \frac{x}{\sqrt{5 - |x + 3|}}
\]

The task is to find the domain of this function, i.e., the values of \(x\) for which the function is defined.

### Conditions for the domain:

1. **The expression inside the square root must be non-negative**: 
   \[
   5 - |x + 3| \geq 0
   \]
   This ensures that the square root is real (since we can't take the square root of a negative number in the set of real numbers).

2. **The denominator cannot be zero**: 
   \[
   \sqrt{5 - |x + 3|} \neq 0
   \]
   This avoids division by zero.

Let's solve these conditions.

### Step 1: Solve \( 5 - |x + 3| \geq 0 \)

This inequality means:
\[
|x + 3| \leq 5
\]
We now solve this absolute value inequality.

#### Remove the absolute value:

\[
-5 \leq x + 3 \leq 5
\]

#### Solve for \(x\):

Subtract 3 from all sides:
\[
-8 \leq x \leq 2
\]

Thus, \(x\) must lie between \(-8\) and \(2\) inclusive.

### Step 2: Exclude the points where the denominator is zero

We need to ensure that \(5 - |x + 3| \neq 0\). This happens when:

\[
5 - |x + 3| = 0
\]

\[
|x + 3| = 5
\]

Solve this for \(x\):

\[
x + 3 = 5 \quad \text{or} \quad x + 3 = -5
\]

\[
x = 2 \quad \text{or} \quad x = -8
\]

These are the points where the denominator would be zero, so we must exclude \(x = 2\) and \(x = -8\) from the domain.

### Final Answer: 
The domain of the function is:

\[
x \in (-8, 2)
\]

This is the interval where the function is defined and real.

Let me know if you'd like a more detailed explanation or have any other questions!

---

### Relative Questions:
1. How do we handle absolute value inequalities in general?
2. Why can't we have negative values inside a square root in real functions?
3. What happens if the denominator of a function is zero?
4. Can a function be defined at the endpoints of an interval?
5. How would the domain change if the square root wasn't in the denominator?

### Tip:
Always check for conditions like square roots and denominators when finding a function's domain to avoid undefined operations.

Raskite funkcijos f(x) = x / sqrt(5 - |x + 3|) apibrėžimo sritį.

This problem involves finding the domain of the function f(x) = x / sqrt(5 - |x + 3|). Explore key mathematical concepts such as absolute value inequalities, square roots, and domain restrictions. Suitable for Grades 9-12.

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