## Related questions with answers

You are driving a car when a deer suddenly darts across the road in front of you. Your brain registers the emergency and sends a signal to your foot to hit the brake. The car travels a reaction distance D, in feet, during this time, where D is a function of the speed r, in miles per hour, that the car is traveling when you see the deer, given by $D(r)=\frac{11 r+5}{10}.$ Find the inverse and explain what it represents. Is the inverse a function?

Solution

Verified$\begin{align} D(r)&=\dfrac{11r+5}{10}\\ D&=\dfrac{11r+5}{10}\\ 10D&=11r+5\\ 10D-5 &=11r\\ r&=\dfrac{10D-5}{11}\\ r(D)&=\dfrac{10D-5}{11} \end{align}$

The last line is the inverse. The inverse means given a specific value of $D$, in feet, we can find the speed $r$, in mph, using the equation $r(D) =\dfrac{10D-5}{11}$. The inverse is a function since for each value of $D$, there is exactly one value of $r$, and vice versa.

To find the inverse of $D(r)$, i.e., $D^{-1}(r)$:

(1) Given

(2) Write $D(r)$ just as $D$.

(3)$-$(5) Solve for $r$.

(6) Write $r$ as $r(D)$. This is the inverse.

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