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.