To find the domain of the function \( f(x) = \sqrt{\log_{10}\left(\frac{5x - x^2}{4}\right)} \), we need to ensure that the expression inside the square root is non-negative and that the argument of the logarithm is positive. ### Step-by-Step Solution: 1. **Condition for the logarithm**: The argument of the logarithm must be positive: \[ \frac{5x - x^2}{4} > 0 \] This simplifies to: \[ 5x - x^2 > 0 \] Rearranging gives: \[ x(5 - x) > 0 \] 2. **Finding the roots**: The roots of the equation \( x(5 - x) = 0 \) are \( x = 0 \) and \( x = 5 \). 3. **Testing intervals**: We need to test the intervals determined by these roots: \( (-\infty, 0) \), \( (0, 5) \), and \( (5, \infty) \). - For \( x < 0 \): Choose \( x = -1 \): \[ -1(5 - (-1)) = -1(6) < 0 \quad \text{(not valid)} \] - For \( 0 < x < 5 \): Choose \( x = 1 \): \[ 1(5 - 1) = 1(4) > 0 \quad \text{(valid)} \] - For \( x > 5 \): Choose \( x = 6 \): \[ 6(5 - 6) = 6(-1) < 0 \quad \text{(not valid)} \] Thus, the solution to \( x(5 - x) > 0 \) is: \[ 0 < x < 5 \quad \text{or} \quad (0, 5) \] 4. **Condition for the square root**: The expression inside the square root must be non-negative: \[ \log_{10}\left(\frac{5x - x^2}{4}\right) \geq 0 \] This implies: \[ \frac{5x - x^2}{4} \geq 1 \] Multiplying both sides by 4 gives: \[ 5x - x^2 \geq 4 \] Rearranging gives: \[ -x^2 + 5x - 4 \geq 0 \] or \[ x^2 - 5x + 4 \leq 0 \] 5. **Finding the roots of the quadratic**: The roots of \( x^2 - 5x + 4 = 0 \) can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{5 \pm \sqrt{25 - 16}}{2} = \frac{5 \pm 3}{2} \] This gives: \[ x = 4 \quad \text{and} \quad x = 1 \] 6. **Testing intervals for the quadratic inequality**: We need to test the intervals determined by the roots \( 1 \) and \( 4 \): \( (-\infty, 1) \), \( (1, 4) \), and \( (4, \infty) \). - For \( x < 1 \): Choose \( x = 0 \): \[ 0^2 - 5(0) + 4 = 4 > 0 \quad \text{(not valid)} \] - For \( 1 < x < 4 \): Choose \( x = 2 \): \[ 2^2 - 5(2) + 4 = 4 - 10 + 4 = -2 \leq 0 \quad \text{(valid)} \] - For \( x > 4 \): Choose \( x = 5 \): \[ 5^2 - 5(5) + 4 = 25 - 25 + 4 = 4 > 0 \quad \text{(not valid)} \] Thus, the solution to \( x^2 - 5x + 4 \leq 0 \) is: \[ 1 \leq x \leq 4 \quad \text{or} \quad [1, 4] \] 7. **Finding the intersection of the two conditions**: The domain of \( f(x) \) is the intersection of \( (0, 5) \) and \( [1, 4] \): \[ (0, 5) \cap [1, 4] = [1, 4] \] ### Final Answer: The domain of \( f(x) = \sqrt{\log_{10}\left(\frac{5x - x^2}{4}\right)} \) is: \[ \boxed{[1, 4]} \]

The domain of ` sqrt(log_10((5x-x^2)/4))` is

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