No. The inverse of the secant is called the arc-secant. The relation between the secant and the cosecant is similar to the relation between the sine and the cosine - they are somehow related, but they are not inverse functions.
The secant is the reciprocal of the cosine (sec x = 1 / cos x).
The cosecant is the reciprocal of the sine (cos x = 1 / sin x).
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The inverse of the inverse is the original function, so that the product of the two functions is equivalent to the identity function on the appropriate domain. The domain of a function is the range of the inverse function. The range of a function is the domain of the inverse function.
The original function's RANGE becomes the inverse function's domain.
The inverse of the cubic function is the cube root function.
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No, an function only contains a certain amount of vertices; leaving a logarithmic function to NOT be the inverse of an exponential function.