To analyze the function $y = \ln\left(x^2 - 3x + 2\right) + \sqrt{-x^2 + 4x}$, we need to consider the domain and behavior of each component separately.

1. Domain of the Function

For the function to be well-defined:

• The natural logarithm $\ln(z)$ requires that $z > 0$.
• The square root $\sqrt{z}$ requires that $z \geq 0$.

For the logarithmic term $\ln(x^2 - 3x + 2)$:

• The argument $x^2 - 3x + 2$ must be greater than zero.
• $x^2 - 3x + 2$ factors as: $x^2 - 3x + 2 = (x - 1)(x - 2)$
• The product $(x - 1)(x - 2) > 0$, which occurs when:
  • $x < 1$, or
  • $x > 2$.

For the square root term $\sqrt{-x^2 + 4x}$:

• The argument $-x^2 + 4x$ must be non-negative.
• We can factor it as: $-x^2 + 4x = - (x^2 - 4x) = -x(x - 4)$
• Therefore, $-x(x - 4) \geq 0$, which occurs when:
  • $0 \leq x \leq 4$.

2. Intersection of Domains

The overall domain is the intersection of the two domains:

• From the logarithm: $x < 1$ or $x > 2$.
• From the square root: $0 \leq x \leq 4$.

Thus, the valid intersection is $0 \leq x < 1$ or $2 < x \leq 4$.

3. Final Domain

So, the domain of the function $y$ is: $\boxed{[0, 1) \cup (2, 4]}$

Additional Analysis

If you need further details, we could explore the behavior of $y$ (e.g., increasing/decreasing intervals, critical points, etc.).

Would you like a deeper analysis of any particular aspect of the function?

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Here are 5 related questions:

1. What are the critical points of the function $y = \ln(x^2 - 3x + 2) + \sqrt{-x^2 + 4x}$?
2. How would the graph of this function behave near the domain boundaries?
3. Can you find the intervals where the function is increasing or decreasing?
4. How would you find the maximum and minimum values of this function on its domain?
5. What happens to the function as $x$ approaches 1 or 2 from within the domain?

Tip: Always factor quadratic expressions to help determine domains easily.