1. One-to-One Functions and Inverse Functions#6
One-to-One Functions and Inverse Functions

2. Every x is paired with one y every y is paired with one x.
One-to-One Function: each x in the domain of f is paired exactly one y in the range and no y in the range is the image of more than one x in the domain.
Every x is paired with one y
and
every y is paired with one x.
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 y3 x3 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
Not a
function y3
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 cubic function is one-to-one.

7. for every x in the domain of f and f(f -1(x)) = 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. How do I know?
Find the inverse of
The function is one-to-one. How do I know?
1. Interchange variables.
2. Solve for y
3. Verify.

14. Find the domain and range of f(x) and g(x).

15. If a function is not one-to-one, then its inverse is not a function
If a function is not one-to-one, then its inverse is not a function. However, we may restrict the domain of such a function so that its inverse is a function that is one-to-one.
Example: Find the inverse of f(x) = x2 if x ≥0.

16. Example #1 Is the following function one-to-one?
If so, find its inverse.

17. Example #2 Is the following function one-to-one?
If so, find its inverse.