Finding "own" price elasticity of demand

Question

Only two groups buy the Dripfix, abbreviated as good $D$. Group 1's demand function for the Dripfix is $f_1 (p_D)=300-p_D$, where $p_D$ is the price of Dripfix. Group 2's demand function is $f_2(p_D)=120-p_D$.

1. What is each group's "own" price elasticity of demand?
2. What is the equation for the market demand?
3. At what price $p_D$ is the market elasticity $-1$?

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Here is my work so far:

1. Group 1's price elasticity is $\epsilon_1=-\frac{p_D}{300-p_D}$. Group 2's is $\epsilon_2=-\frac{p_D}{120-p_D}$.
2. The market demand is $420-2p_D$ for $p_D \leq 120$ and $300-p_D$ for $p_D > 120$.

Are these two parts correct so far? If so, how do go about determining the market elasticity when the market demand is made up of two components?

Answer 1

Your work for 1 and 2 is correct.

For 3, look at each segment individually. There are two points which have an elasticity of $-1$, one on each segment.