1. One-to-One Functions; Inverse Function

2. A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain.
A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

3. x1 y1 x1 y1 x2 y2 x2 x3 x3 y3 y3 One-to-one function NOT One-to-one
Domain
Range
Domain
Range
One-to-one
function
NOT One-to-one
function x1 y1 y2 x3 y3
Not a
function
Domain
Range

4. M: Mother Function is NOT one-one
Joe
Samantha
Anna
Ian
Chelsea
George
Laura
Julie
Hilary
Barbara
Sue
Humans
Mothers

5. S: Social Security function IS one-one
Joe
Samantha
Anna
Ian
Chelsea
George
Americans
SSN

6. Is the function f below one – one?10 11 12 13 14 15 16 1 2 3 4 5 6 7

7. Theorem Horizontal Line Test
If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

8. Use the graph to determine whether the function
is one-to-one.
Not one-to-one.

9. Use the graph to determine whether the function is one-to-one.

10. The inverse of a one-one function is obtained by switching the role of x and y

11. The inverse of the social security function
Joe
Samantha
Anna
Ian
Chelsea
George
SSN
Americans

12. Let and
Find

13. g is the inverse of f.

14. Let f denote a one-to-one function y = f(x)
Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1 , is a function such that f -1(f( x )) = x for every x in the domain of f and f(f -1(x))=x for every x in the domain of f -1. .

15. Domain of f
Range of f

16. Theorem
The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

17. y = x
(0, 2)
(2, 0)

18. Finding the inverse of a 1-1 function
Step1: Write the equation in the form
Step2: Interchange x and y.
Step 3: Solve for y.
Step 4: Write for y.

19. Find the inverse of Step1: Step2: Interchange x and y
Step 3: Solve for y