Inverse functions and temperature
Recall that if the temperature in degrees Celsius is
$C$, the temperature in degrees Fahrenheit is given by $$F=9/5 C + 32$$.
If $$F=9/5 C + 32$$ and $C=20$, what is $F$?
| If the temperature is 20 degrees Celsius, what is the temperature
in degrees Fahrenheit? (Use your answer to the previous question.)
| degrees Fahrenheit |
| If the temperature is $-5$ degrees Celsius, what is the temperature
in degrees Fahrenheit? (Use scratch paper.)
| degrees Fahrenheit |
One way to think of this formula is as a function $T(C)$ which transforms degrees Celsius
into degrees Fahrenheit. That is, a temperature of $C$ degrees Celsius means the same thing as
a temperature of $$T(C)=9/5 C + 32$$ degrees Fahrenheit. So you have just computed the
values of $T(20)$ and $T(-5)$.
Remember that if you have a linear function like $T(C)$, you can find a formula for its
inverse, $T^{-1}$, by solving the equation $T(C)=F$ for $C$. Find this
formula for $C=T^{-1}(F)$. The function $T^{-1}$ will transform “temperature in degrees
Fahrenheit” into “temperature in degrees Celsius.”
In temperature-question-1, we found a formula for
$T^{-1}(F)$:
$$
T^{-1}(F)=5/9 F - 160/9
$$
Just as the function $T(C)$ changes Celsius temperatures into Fahrenheit temperatures, the
function $T^{-1}(F)$ changes Fahrenheit temperatures into Celsius temperatures.
Use this formula for $T^{-1}(F)$ to find $T^{-1}(41)$.
| If the temperature is 41 degrees Fahrenheit, what is the
temperature in degrees Celsius?
| degrees Celsius |
| If the temperature is 23 degrees Fahrenheit, what is the temperature
in degrees Celsius? (Use scratch paper.)
| degrees Celsius |
In temperature-question-1, you found that $T(-5)=23$. Using this and
your previous answer, compute the value of $T^{-1} ∘ T(-5)$.
What is $T ∘ T^{-1}(23)$?
When we talked about Fahrenheit and Celsius temperatures near the beginning of this course,
you needed to solve a different equation to find the temperature in degrees Celsius for each
Fahrenheit temperature you were given. By using inverse functions, we’ve discovered a single
formula for $T^{-1}(F)$ that you can use for any temperature!
Vending machines and shifting/stretching functions
A Mathscribe Munchies vending machine costs \$1000. Every year you own it, you
will make \$200 in profit. So, if you own one, the total profit you will make after $t$ years
is:
$$
\cl"red"{P(t)=200t-1000}
$$
This function $P(t)$ is graphed on the grid to the left. (When your
total profit is negative, that means that you still have less money than when you started.)
| How much of a profit will you make after 2 years?
| |
| How much of a profit will you make after 6 years?
| |
| How long will it take for you to break even
(that is, for your profit to stop being negative)? (Use the graph of $P(t)$,
or solve the equation $P(t)=0$ on scratch paper.)
| years |
The Mathscribe Munchies company decides to offer a \$200 rebate when you buy a vending
machine from them. This means that, if you buy the machine, you’ll have an extra \$200 to
start with, so your total profit after $t$ years will be $P(t)+200$.
Click to see a graph of $P(t)+200$
along with the graph of $P(t)$.
| If you buy a vending machine and get a rebate, how much of a profit
will you make after 2 years?
| |
| How much of a profit will you make after 6 years?
| |
| How long will it take for you to break even?
| years |
| Is the blue graph of
$P(t)+200$ above or below the
red graph of $P(t)$?
| |
| Will you make more money or less money if you buy the
machine with the rebate?
| |
You now would like to think about buying three Mathscribe Munchies machines
instead of one. If you do this, your profit after $t$ years will be
$3P(t)$. This is graphed on the grid to the left, along with the
graph of $\cl"red"{P(t)}$. (The Mathscribe Munchies company no longer offers rebates on its
vending machines.)
| If you buy three vending machines, how much of a profit will you make
after 2 years?
| |
| How much of a profit will you make after 6 years?
| |
| If you buy three machines, will you break even sooner,
later, or at the same time as if you buy one machine?
| |
| When $t$ is large (after you break even), is the
green graph above or below the
red graph?
| |
| When $t$ is small (before you break even), is the
green graph above or below the
red graph?
| |
| If you end up running your vending machines for a long time, is it
better to own one vending machine or three vending machines?
| |
| If your vending machines stop working before you break even, is it
better to own one vending machine or three vending machines?
| |
Grains of rice on a chessboard
There is a story that’s at least a thousand years old, about a wise man who served
his king well for many years. When the king asked him what he wanted as a reward for his
service, he said:
“On the first square of this chessboard, I would like you to put a grain of rice. On the next
square, put two grains of rice. On the square after that, put four grains of rice. On each
square, put twice as many grains of rice as you did on the square before. Once you have done
this for every square on the chessboard, I will take that rice as my reward.”
| If the third square is to have 4 grains of rice, and each square is
to have twice as many grains of rice as the square before it, how many grains of rice will there
be on the fourth square?
| |
| How many grains of rice will there be on the fifth square?
| |
We’ll look at the sequence $C(n)$ whose first few terms are
$$
1,2,4,8,16,...
$$
$C(n)$ is the number of grains of rice that the king is supposed to put on the $n$th square
of the chessboard.
| Each number in the sequence $C(n)$ is twice as large as the number
before it. Is this sequence arithmetic, geometric, or neither?
| |
| What is the common ratio of this sequence?
| |
Because $C(n)$ is a geometric sequence with common ratio $2$, there must be a
formula for $C(n)$ which looks like $C(n)=a ⋅ 2^n$ for some number $a$.
This formula tells us that $C(1)=a ⋅ 2^1 = 2a$. Also, there should be one grain of rice on
the first square, so $C(1)=1$. If $C(1)=2a$ and $C(1)=1$, then $2a=1$.
| What is the solution to the equation $2a=1$?
| |
| What is a formula for $C(n)$?
| |
| Using this formula and a calculator, compute $C(30)$.
| |
| There are about 60 million (60,000,000) grains of rice in a ton. About
how many tons of rice is the king supposed to put on the 30th square of the chessboard? (Round
to the nearest whole ton.)
| tons |
| Remembering that the amount of rice on each square is double the
amount on the square before it, about how many tons of rice is the king supposed to put on the
31st square of the chessboard?
| tons |
| A chessboard has 64 squares on it. Is the king likely to be able to
fulfill the wise man’s request?
| |