To determine the domain of the function f(x)=log10(x3−x), we need to ensure that the expression inside the square root is non-negative and that the argument of the logarithm is positive. Let's break this down step by step.1. Argument of the Logarithm Must Be Positive:The argument of the logarithm, x3−x, must be positive. This gives us the inequality:x3−x>0To solve this inequality, we need to determine where the expression x3−x is positive. The critical points are x=0 and x=3. We can test the intervals determined by these points: (−∞,0), (0,3), and (3,∞).
For x∈(0,3), both 3−x and x are positive, so x3−x>0.
For x∈(−∞,0) or x∈(3,∞), the expression x3−x is not positive.Therefore, the inequality x3−x>0 is satisfied for 0<x<3.
2. Expression Inside the Square Root Must Be Non-Negative:The expression inside the square root, log10(x3−x), must be non-negative. This means:log10(x3−x)≥0Since the logarithm base 10 of a number is non-negative if and only if the number is at least 1, we have:x3−x≥1Simplifying this inequality:x3−x−1≥0⇒x3−x−x≥0⇒x3−2x≥0The critical points are x=0 and x=23. We test the intervals determined by these points: (−∞,0), (0,23), and (23,∞).
For x∈(0,23), both 3−2x and x are positive, so x3−2x>0.
For x∈(23,∞), 3−2x is negative, so x3−2x<0.Therefore, the inequality x3−2x≥0 is satisfied for 0<x≤23.
Combining the two conditions, we get that the domain of f(x) is:0<x≤23Thus, the domain of f is (0,23].The correct option is B.