The function $f(x) = \sqrt{|3x - 8| - 5}$ is defined under certain conditions. Let’s analyze it in detail.

Step 1: Analyze the inner expression $|3x - 8| - 5$

1. The square root function $\sqrt{y}$ is only defined for $y \geq 0$. Hence, we need: $|3x - 8| - 5 \geq 0.$

2. Rearranging, we get: $|3x - 8| \geq 5.$

3. The absolute value inequality $|u| \geq a$ splits into two cases: $3x - 8 \geq 5 \quad \text{or} \quad 3x - 8 \leq -5.$

Step 2: Solve each inequality

Case 1: $3x - 8 \geq 5$ $3x \geq 13 \implies x \geq \frac{13}{3}.$

Case 2: $3x - 8 \leq -5$ $3x \leq 3 \implies x \leq 1.$

Step 3: Combine the results

The domain of $f(x)$ is the union of the intervals where the inequality holds: $x \in (-\infty, 1] \cup \left[\frac{13}{3}, \infty\right).$

Final Function Domain

The domain of $f(x)$ is: $x \in (-\infty, 1] \cup \left[\frac{13}{3}, \infty\right).$

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Graphical Behavior

• The function has a square root, so it is always non-negative.
• The absolute value introduces a piecewise behavior around $x = \frac{8}{3}$, where $|3x - 8|$ transitions between $3x - 8$ and $-(3x - 8)$.

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