Final answer:
The domain of the function f(x) is the interval where x is between 0 and 1, excluding the endpoint 1, due to the constraints of the arc sine and arc cosine functions and the requirement for the argument of the logarithm to be positive.
Step-by-step explanation:
When finding the domain of the given function f(x) = 1/√sin-1x log10(cos⁻¹x), we must consider the constraints on both arc sine (sin-1) and arc cosine (cos⁻¹) functions, as well as the requirement for the argument of the logarithm to be positive.
The arc sine function, sin-1x, is defined for x ∈ [-1, 1]. However, since it's under a square root in the function, we also require that sin-1x must be non-negative, thereby restricting x to be in [0,1].
Moreover, the arc cosine function, cos⁻¹x, is also defined for x ∈ [-1, 1], but the log10 function requires its argument to be strictly positive, so cos⁻¹x cannot equal 1 (which would make log10(0) undefined). Given that cos⁻¹x decreases from 1 to 0 as x increases from 0 to 1, we require x to be exclusively less than 1 for cos⁻¹x to be positive.
Combining both restrictions, the only interval that satisfies both conditions is when x is between 0 and 1, but not including 1. Hence the domain is x ∈ (0, 1).