A family of pdfs or pmfs is called an exponential family if
$$f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$$
so, I wonder if this distribution belongs to the exponential family:
$$f(x|\theta) = exp(\theta-x)$$
My guess is no because we cannot write it as above, am I correct?