Check for Understanding: Inverse Function Machine

Question

Max challenges Maria to a number game. He asks her to pick a number between 1 and 10, confident that he can tell her this original number after she completes the following steps. Maria picks a number and Max then tells her to do the following operations: 

1. Add four
2. Multiply by 3
3. Subtract 10

Max then asks Maria for her new value. Maria says "29." Max replies:

"OK! Your original number is _________!" 

Answer 1

As with most algebra problems, the key to success is setting up the problem correctly.

Here, Maria picks an input value which we will call x, and after carrying out Max's instructions, she obtains an output value, which we will call y. Since the operations give a single output for each input, we can think of Max's instructions as a function y = f(x). Step-by-step,

1. Add four: x + 4
2. Multiply by three: 3(x + 4)
3. Subtract ten: 3(x + 4) − 10

So we see that the function f(x) is given by f(x) = 3(x + 4) − 10. What Max is trying to do is to find the x-value, given the output y. In other words, Max is trying to find the inverse function f^-1(y) = x. Thus, he is taking this equation and solving algebraically for x in terms of y. To do this, we just reverse the order of operations:

1. Add ten: y + 10 = 3(x + 4)
2. Divide by three: (y + 10)/3 = x + 4
3. Subtract four: (y + 10)/3 − 4 = x.

We've found the inverse! It is f^-1(y) = (y + 10)/3 − 4. To find the number Maria started with, Max needs to simply find f^-1(29) using this formula. Plugging in y = 29, we see that Maria's starting number was in fact 9.