1. 6.1 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. 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.

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

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

7. 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. .

8. Domain of f
Range of f

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

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

11. The function is one-to-one.
Find the inverse of
The function is one-to-one.
Interchange variables.
Solve for y.

12. Check.