1.7 Inverse Functions (continued)

Properties of an inverse function

Summary

Compute $g(3)$ and write a complete sentence to explain its meaning in the given context, including units.

Compute the average rate of change of $g$ on the time interval $[3,5]$ and write two careful complete sentences to explain the meaning of this value in the context of the problem, including units. Explicitly address what the value you compute tells you about how rain is falling over a certain time interval, and what you should expect as time goes on.

Plot the function $y = g(t)$ using a computational device. On the domain $[0,10]$, what is the corresponding range of $g$? Why does the function $g$ have an inverse function?

Determine $g^{-1} \left( \frac{9}{5} \right)$ and write a complete sentence to explain its meaning in the given context.

According to the model $g$, is there ever a time during the storm that the rain falls at a rate of exactly $1$ centimeter per hour? Why or why not? Provide an algebraic justification for your answer.

Consider the functions $p$ and $q$ whose graphs are given by Figure

• Compute each of the following values exactly, or explain why they are not defined: $p^{-1}(2.5)$, $p^{-1}(-2)$, $p^{-1}(0)$, and $q^{-1}(2)$. - From your work in (a), you know that the point $(2.5, -3.5)$ lies on the graph of $p^{-1}$. In addition to the other two points you know from $(a)$, find three additional points that lie on the graph of $p^{-1}$. - On Figure, plot the $6$ points you have determined in (a) and (b) that lie on the graph of $y = p^{-1}(x)$. Then, sketch the complete graph of $y = p^{-1}(x)$. How are the graphs of $p$ and $p^{-1}$ related to each other?

Consider an inverted conical tank that is being filled with water. The tank's radius is $2$ m and its depth is $4$ m. Suppose the tank is initially empty and is being filled in such a way that the height of the water is always rising at a rate of $0.25$ meters per minute.

• Explain why the height, $h$, of the water can be viewed as a function of $t$ according to the formula $h= f(t) = 0.25t$. - At what time is the water in the tank $2.5$ m deep? At what time is the tank completely full? - Suppose we think of the volume, $V$, of water in the tank as a function of $t$ and name the function $V = g(t)$. Do you expect that the function $g$ has an inverse function? Why or why not? - Recall that the volume of a cone of radius $r$ and height $h$ is $V = \frac{\pi}{3} r^2 h$. Due to the shape of the tank, similar triangles tell us that $r$ and $h$ satisfy the proportion $r = \frac{1}{2}h$, and thus

. Use the fact that $h = f(t) = 0.25t$ along with Equation to find a formula for $V = g(t)$. Sketch a plot of $V = g(t)$ on the blank axes provided in Figure. Write at least one sentence to explain why $V = g(t)$ has the shape that it does. - Take the formula for $V = g(t)$ that you determined in (d) and solve for $t$ to determine a formula for $t = g^{-1}(V)$. What is the meaning of the formula you find? - Find the exact time that there is $\frac{8}{3}\pi$ cubic meters of volume in the tank.

Recall that in Activity 1.6.2, we showed that Celsius temperature is a function of the number of chirps per minute from a snowy tree cricket according to the formula

.

• What familiar type of function is $H$? Why must $H$ have an inverse function? - Determine an algebraic formula for $N = H^{-1}(C)$. Clearly show your work and thinking. - What is the meaning of the statement $72 = H^{-1}\left(\frac{130}{9}\right)$? - Determine the average rate of change of $H$ on the interval $[40,50]$. Write a complete sentence to explain the meaning of the value you find, including units on the value. Explain clearly how this number describes how the temperature is changing. - Determine the average rate of change of $H^{-1}$ on the interval $[15,20]$. Write a complete sentence to explain the meaning of the value you find, including units on the value. Explain clearly how this number describes how the number of chirps per minute is changing.