1. Lecture 37 Section 7.2 Tue, Apr 3, 2007
Inverse Functions
Lecture 37
Section 7.2
Tue, Apr 3, 2007

2. Inverse Relations Let R be a relation from A to B.
The inverse relation of R is the relation R–1 from B to A defined by the property that (x, y)  R–1 if and only if (y, x)  R.
If a function f : A  B is a one-to-one correspondence, then it has an inverse function f -1 : B  A such that if f(x) = y, then f -1(y) = x.

3. Example: Inverse Relation
Let f : R  R by f(x) = 2x.
Describe f –1.

4. Example: Inverse Relation
Let g : R*  R* by g(x) = 1/x.
Describe g–1.

5. Example: Inverse Relation
Let k : R  R by k(x) = x.
Describe k–1.

6. Example: Inverse Functions
Let A = R – {1/3}.
Let B = R – {2/3}.
Define f : A  B by f(x) = 2x/(3x – 1).
Find f –1.
Let y = 2x/(3x – 1).
Swap x and y: x = 2y/(3y – 1).
Solve for y: y = x/(3x – 2).
Therefore, f –1(x) = x/(3x – 2).

7. Example: Inverse Relation
Let A = R and B = R.
Let j : A  B by j(x) = (3x – 1)/(x + 1).
Find j -1.
What values must be deleted from A and B to make j a one-to-one correspondence?
Verify that the modified j is one-to-one and onto.

8. Inverse Relations and the Basic Properties
A relation R has the first basic property if and only if R–1 has the third basic property.
x  A,  at least one y  B, (x, y)  R.
y  B,  at least one x  A, (x, y)  R.

9. Inverse Relations and the Basic Properties
A relation R has the second basic property if and only if R–1 has the fourth basic property.
x  A,  at most one y  B, (x, y)  R.
y  B,  at most one x  A, (x, y)  R.

10. Inverse Functions
Theorem: The inverse of a function is itself a function if and only if the function is a one-to-one correspondence.
Corollary: If f is a one-to-one correspondence, then f –1 is a one-to-one correspondence.
The inverse of a function is, in general, a relation, but not a function.

11. Q and Z Theorem: There is a one-to-one correspondence from Z to Q.
Proof:
Consider only rationals in reduced form.
Arrange the positive rationals in order
First by the sum of numerator and denominator.
Then, within groups, by numerator.

12. Q and Z The sequence is 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, …
The first group: 1/1
The second group: 1/2, 2/1
The third group: 1/3, 3/1
The fourth group: 1/4, 2/3, 3/2, 4/1
Etc.
The sequence is 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, …

13. Q and Z Let f : Z  Q be the function that
Maps the positive integer n to the nth rational in this list.
Maps the negative integer -n to the negative of the rational that n maps to.
Maps 0 to 0.
This is a one-to-one correspondence.

14. Q and Z
What is f(20)?
What is f –1(4/5)?