Question

The domain of the function $f(x) = \sqrt{\log_{10} \left(\frac{5x - x^2}{4}\right)}$ is:

• A. $[1, 4]$
• B. $(1, 4)$
• C. $[0, 5]$
• D. $(0, 5)$

Hint:

Test boundaries! At $x=1$, we get $\log_{10}(1) = 0$, which is valid under a square root, meaning $1$ must be included (closed interval). At $x=0$, the logarithm argument is $0$, which is undefined, meaning $0$ must be excluded.

Answer 1

The Correct Option is A

Step 1: Two rules to keep things real.
A square root needs the inside to be zero or more. A log needs its inside to be strictly more than zero. Both must hold at the same time.

Step 2: First condition (log is defined).
The inside of the log must be positive: \[ \frac{5x - x^2}{4} > 0 \implies x(5-x) > 0 \] This is true between the roots, so $x \in (0,5)$.

Step 3: Second condition (square root is real).
The inside of the root is the log itself, so we need \[ \log_{10}\left(\frac{5x - x^2}{4}\right) \ge 0 \]

Step 4: Remove the log.
A log being $\ge 0$ means its inside is $\ge 10^0 = 1$. So \[ \frac{5x - x^2}{4} \ge 1 \implies 5x - x^2 \ge 4 \]

Step 5: Solve the new quadratic.
Rearrange: \[ x^2 - 5x + 4 \le 0 \implies (x-1)(x-4) \le 0 \] This holds for $x \in [1,4]$.

Step 6: Take the common part.
The point must satisfy both, so take the overlap of $(0,5)$ and $[1,4]$. The smaller one wins. \[ \boxed{ x \in [1, 4] } \]