If f is a one-to-one function and f(0) = 3, f(1) = 1, and f(2) = 2, find the following:

Question

(a) f^-1(1)

f^-1(1)=?

(b) f^-1(2)

f^-1(2)=?

Answer 1

So assuming f is 'one-to-one' means each number gets assigned to its own special number. No other number will be assigned to that special number. In this case that means f assigns 0 to 3 and 0 is the only number assigned to 3. The same is true for 1 being sent to 1 and 2 being sent to 2 by f.

Since each number in goes to one special number out we can create a new function f^-1 which reverses these assignments. I think this example is small enough that a diagram can be quite helpful. We can picture the assignments for f like this:

0 → 3

1 → 1

2 → 2

(I read this like 'f sends 0 to 3, 1 to 1, and 2 to 2'.)

We get f^-1 by just reversing the arrows (the -1 is supposed to refer to the function going 'backward' compared to f):

0 ← 3

1 ← 1

2 ← 2

From here we can see all of the assignments made by f^-1:

1. f^-1(3) = 0
2. f^-1(1) = 1
3. f^-1(2) = 2

Hopefully that helps to make the concept clearer here.

Answer 2

f_inverse(1) = 1

f_inverse(2)=2