1. 4.1 Inverse Functions

2. 5.1 One-to-One Functions
Only functions that are one-to-one have inverses.
A function f is a one-to-one function if, for elements a and b from the domain of f,
a  b implies f (a)  f (b).

3. 5.1 One-to-One Functions
Example Decide whether the function is one-to-one.
(a) (b)
Solution
(a) For this function, two different x-values produce two different y-values.
(b) If we choose a = 3 and b = –3, then 3  –3, but

4. 5.1 The Horizontal Line Test
Example Use the horizontal line test to determine
whether the graphs are graphs of one-to-one functions.
(a) (b)
If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one.
Not one-to-one
One-to-one

5. 5.1 Inverse Functions
Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if
Example
are inverse functions of each other.

6. 5.1 Finding an Equation for the Inverse Function
Notation for the inverse function f -1 is read
“f-inverse”
Finding the Equation of the Inverse of y = f(x)
1. Interchange x and y.
2. Solve for y.
3. Replace y with f -1(x).
Any restrictions on x and y should be considered.

7. 5.1 Example of Finding f -1(x)
Example Find the inverse, if it exists, of
Solution
Write f (x) = y.
Interchange x and y.
Solve for y.
Replace y with f -1(x).

8. 5.1 The Graph of f -1(x)
f and f -1(x) are inverse functions, and f (a) = b for real numbers a and b. Then f -1(b) = a.
If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1.
If a function is one-to-one, the graph of its inverse f -1(x) is a reflection of the graph of f across the line y = x.

9. 5.1 Finding the Inverse of a Function with a Restricted Domain
Example Let
Solution Notice that the domain of f is restricted
to [–5,), and its range is [0, ). It is one-to-one and
thus has an inverse.
The range of f is the domain of f -1, so its inverse is

10. 5.1 Important Facts About Inverses
If f is one-to-one, then f -1 exists.
The domain of f is the range of f -1, and the range of f is the domain of f -1.
If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1, so the graphs of f and f -1 are reflections of each other across the line y = x.

11. 5.1 Application of Inverse Functions
Example Use the one-to-one function f (x) = 3x + 1 and the
numerical values in the table to code the message BE VERY CAREFUL.
A 1 F 6 K 11 P 16 U 21
B 2 G 7 L 12 Q 17 V 22
C 3 H 8 M 13 R 18 W 23
D 4 I 9 N 14 S 19 X 24
E 5 J O 15 T 20 Y 25
Z 26
Solution BE VERY CAREFUL would be encoded as
because B corresponds to 2, and f (2) = 3(2) + 1 = 7,
and so on.