College Algebra, Chapter 3, 3.7, Section 3.7, Problem 78

The amount of a commodity that is sold is called the demand for the commodity. The equation $D(p) = -3p + 150$ represents the demand for a certain commodity as a function of price.

a.) Find $D^{-1}$. What does $D^{-1}$ represent?

b.) Find $D^{-1} (30)$. What does it represent?

a.) To find $D^{-1}$, we set $y = D(p)$.


$\begin{equation} \begin{aligned} y =& - 3p + 150 && \text{Solve for $t$; add $3p$ and subtract $y$} \\ \\ 3p =& 150 - y && \text{Divide by } 3 \\ \\ p =& \frac{150 - y}{3} && \text{Simplify} \\ \\ p =& 50- \frac{y}{3} && \text{Interchange $y$ and $p$} \\ \\ y =& 50 - \frac{p}{3} && \end{aligned} \end{equation}$


Thus, the inverse of $D(p)$ is $\displaystyle D^{-1} (p) = 50 - \frac{p}{3}$.

If $D(p)$ represents the amount of commodity sold, then $D^{-1} (p)$ represents the amount of commodity that has been unsold.

b.)


$\begin{equation} \begin{aligned} D^{-1} (30) =& 50 - \frac{30}{3} \\ \\ =& 50 - 10 \\ \\ =& 40 \end{aligned} \end{equation}$


$D^{-1} (30)$ means that there are 40 items unsold at the price of 30.