In this paper, we explicitly derive unbiased estimators for various functions of the rate parameter of the exponential distribution, including powers of the rate parameter, the $q$th quantile, the $p$th moment, the survival function, the maximum, minimum, probability density function, mean past lifetime, moment generating function, and others. It is also noteworthy that this work corrects a general formula originally proposed by Tate, R. F. (Ann. Math. Statist., 30(2): 341-366, 1959) for constructing unbiased estimators of functions of the exponential distribution's rate parameter in the absence of a location parameter. Additionally, we establish a result demonstrating the asymptotic normality of the proposed unbiased estimators.

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