Find joint density of $X$ and $\exp(-X)$ $X$ is a uniformly distributed random variable in $[-a,a]$.
Find joint density of $X$ and $\exp(-X)$.
Addition: This is in context to the first undergraduate course in probability
I tried to solve it using Y = exp(-X). Then solving for:
P(X=x,Y=y) = P(Y=y|X=x)P(X=x) gives the following result, which I am doubtful about:
P(X=x,Y=y) = f(x) when y = exp(-x) and zero otherwise.
Also, as pointed out in an answer that the Lebesgue measure is actually zero, so there will not exist a density distribution is what I could think of after seeing the distribution I got which is analytically highly discrete.

The pair $\left(X,e^{-X}\right)$ does not have a common density because their common distribution is concentrated on the set
$$\{(x,y)\mid y=e^{-x}, \ -a\le x\le a\}$$
of $0$ Lebesgue measure. There is neither density nor pmf. (The lebesgue measure and the measure belonging to the common distribution of $\left(X,e^{-X}\right)$ are singular.)
