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Inverse Functions

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

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Inverse Functions

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  1. 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 functionf-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 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)?

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