https://r2.erweima.ai/mathbot/image/1726837754257kxc99653.jpg

The equation provided in the image is:

\[
y = \sqrt{\lg\left(\frac{5x - x^2}{4}\right)}
\]

where \( \lg \) represents the logarithm to base 10.

### Steps to analyze the domain of the function:

1. The expression inside the square root must be non-negative because the square root of a negative number is not real. Thus, the logarithmic expression must be non-negative:
   \[
   \lg\left(\frac{5x - x^2}{4}\right) \geq 0
   \]

2. The logarithmic function is defined only for positive arguments, so:
   \[
   \frac{5x - x^2}{4} > 0
   \]

   This simplifies to:
   \[
   5x - x^2 > 0
   \]

   Factoring the quadratic inequality:
   \[
   x(5 - x) > 0
   \]

   This inequality holds when \( 0 < x < 5 \).

3. The logarithmic inequality \( \lg\left(\frac{5x - x^2}{4}\right) \geq 0 \) translates to:
   \[
   \frac{5x - x^2}{4} \geq 1
   \]
   This leads to:
   \[
   5x - x^2 \geq 4
   \]
   Rearranging this gives:
   \[
   x^2 - 5x + 4 \leq 0
   \]
   Factoring this inequality:
   \[
   (x - 1)(x - 4) \leq 0
   \]

   This inequality holds when \( 1 \leq x \leq 4 \).

### Domain:
The domain of the function is the interval \( [1, 4] \), where the expression inside both the square root and logarithm is defined and non-negative.

Would you like to see further steps, or would you like to ask for any clarification?

### Related Questions:
1. How do we solve inequalities involving logarithmic functions?
2. What happens if the logarithmic base is changed?
3. How can the graph of this function be visualized?
4. How does changing the coefficients in the quadratic affect the domain?
5. What if the function had a natural logarithm instead of a base-10 logarithm?

### Tip:
Always check both the domain of the logarithmic function and the condition imposed by the square root when solving these kinds of problems.

Learn how to determine the domain of the function y = sqrt(lg((5x - x^2)/4)) using logarithmic and quadratic inequalities. This problem covers algebraic concepts like domain restrictions, logarithmic properties, and factoring quadratic expressions. Suitable for high school students in Grades 9-12.

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