The expression representing f(g(1))^−1 is 2. Hence the correct option is c.
To find f(g(1))^−1, we first evaluate the inner function g(x) by substituting x=1. This yields g(1)=1^2 =1. Subsequently, we substitute this value into the outer function f(x)=3x−1, where x is replaced by g(1), resulting in f(g(1))=3×1−1=2. Now, to calculate the inverse, f(g(1))^−1, we find the reciprocal of the result, giving us 1/2.
In theoretical terms, the composition of functions involves applying one function's output as the input to another function. In this case, g(1) serves as the input for f(x). The inverse of a function undoes the operation performed by the original function. In summary, f(g(1))^−1 simplifies to 1/2, indicating that the original operation can be reversed to yield this result. The correct expression in the given options is 2, as it represents the reciprocal of f(g(1)), aligning with the theoretical understanding of function composition and inversion. Hence the correct option is c.