Applications of Functions

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?