Example 1

The demand for a product depends on its price. Demand $q$ can be modeled by the demand function $$q(p) = \dfrac{200}{p+4}.$$

Now assume we wish to know the number of products needed for a particular price. Hence, we wish to know the inverse demand function $p(q)$. We can find it by rewriting the demand function:
$$\begin{align} q &= \dfrac{200}{p+4}\\ q \cdot (p+4) &= 200\\ p+4 &= \dfrac{200}{q}\\ p &= \dfrac{200}{q} - 4. \end{align}$$
Hence, the inverse demand function is $$p(q) = \dfrac{200}{q} - 4.$$

We can check whether the functions $p(q)$ and $q(p)$ indeed satisfy the property of the inverse function:
$$\begin{align} q(p(q)) &= \dfrac{200}{p(q)+4} = \dfrac{200}{\tfrac{200}{q} - 4+4} = \dfrac{200}{\tfrac{200}{q}} = 200 \cdot \dfrac{q}{200} = q\\ &\text{en}\\ p(q(p)) &= \dfrac{200}{q(p)} - 4 = \dfrac{200}{\tfrac{200}{p+4}} - 4 = 200 \cdot \dfrac{p+4}{200} - 4 = p+4-4 = p. \end{align}$$