Which of the following functions has a domain of all reals?arcsecantarccosecantarccotangent

To determine which of the given functions has a domain of all real numbers, we need to consider the restrictions on the domain of each function.

1. arcsecant (arcsec): The arcsecant function is the inverse of the secant function. The secant function is defined as 1/cos(x), and its domain is all real numbers except where cos(x) = 0. Since the denominator cannot be zero, the domain of the secant function is all real numbers except where cos(x) = 0. Therefore, the domain of the arcsecant function is the range of the secant function, which is (-∞, -1] U [1, ∞).

2. arccosecant (arccsc): The arccosecant function is the inverse of the cosecant function. The cosecant function is defined as 1/sin(x), and its domain is all real numbers except where sin(x) = 0. Similar to the secant function, the denominator cannot be zero, so the domain of the cosecant function is all real numbers except where sin(x) = 0. Therefore, the domain of the arccosecant function is the range of the cosecant function, which is (-∞, -1] U [1, ∞).

3. arccotangent (arccot): The arccotangent function is the inverse of the cotangent function. The cotangent function is defined as 1/tan(x), and its domain is all real numbers except where tan(x) = 0. Again, the denominator cannot be zero, so the domain of the cotangent function is all real numbers except where tan(x) = 0. Therefore, the domain of the arccotangent function is the range of the cotangent function, which is (-∞, 0) U (0, ∞).

From the above analysis, we can see that the function with a domain of all real numbers is the arccotangent (arccot) function.