The document discusses inverse functions. It defines inverse functions as pairs of one-to-one functions that undo each other. To find the inverse of a function, interchange the x and y variables and solve for y. The graph of an inverse function is a reflection of the original function across the line y = x. Examples are provided to illustrate how to determine if a function is one-to-one, find the inverse function, and relate the graph of a function to its inverse.

1. 4.1 Inverse Functions
Chapter 4 Inverse, Exponential, and Logarithmic
Functions

2. Concepts & Objectives
 Inverse Functions
 Review composition of functions
 Identify 1-1 functions
 Find the inverse of a 1-1 function

3. Composition of Functions
 If f and g are functions, then the composite function, or
composition, of g and f is defined by
 The domain of g  f is the set of all numbers x in the
domain of f such that f x is in the domain of g.
    
 

g f x g f x

4. Composition of Functions (cont.)
 Example: Let and .
c) Write as one function.
  
2 1
f x x  

4
1
g x
x
  
f g x

5. Composition of Functions (cont.)
 Example: Let and .
c) Write as one function.
  
2 1
f x x  

4
1
g x
x
  
f g x
    
 
f g x f g x

4
2 1
1
x
 
 
 

 
8
1
1
x
 

8 1 8 1
1 1 1
x x
x x x
  
  
  
9
1
x
x




6. One-to-One Functions
 In a one-to-one function, each x-value corresponds to
only one y-value, and each y-value corresponds to only
one x-value. In a 1-1 function, neither the x nor the y can
repeat.
 We can also say that f a = f b implies a = b.
A function is a one-to-one function if, for
elements a and b in the domain of f,
a ≠ b implies f a ≠ f b.

7. One-to-One Functions (cont.)
 Example: Decide whether is one-to-one.
   
3 7
f x x

8. One-to-One Functions (cont.)
 Example: Decide whether is one-to-one.
We want to show that f a = f b implies that a = b:
Therefore, f is a one-to-one function.
   
3 7
f x x
   
f a f b

3 7 3 7
a b
    
3 3
a b
  
a b


9. One-to-One Functions (cont.)
 Example: Decide whether is one-to-one.
  2
2
f x x
 

10. One-to-One Functions (cont.)
 Example: Decide whether is one-to-one.
This time, we will try plugging in different values:
Although 3 ≠ ‒3, f 3 does equal f ‒3. This means that
the function is not one-to-one by the definition.
  2
2
f x x
 
  2
3 3 2 11
f   
   
2
3 3 2 11
f     

11. One-to-One Functions (cont.)
 Another way to identify whether a function is one-to-one
is to use the horizontal line test, which says that if any
horizontal line intersects the graph of a function in more
than one point, then the function is not one-to-one.


one-to-one not one-to-one

12. Inverse Functions
 Some pairs of one-to-one functions undo one another.
 For example, if
and
then (for example)
This is true for any value of x. Therefore, f and g are
called inverses of each other.
  8 5
f x x
   
5
8
x
g x


   
8 10 8
10 5 5
f   
 
85 5 80
8
8
8
10
5
g

  

13. Inverse Functions (cont.)
 More formally:
Let f be a one-to-one function. Then g is the inverse
function of f if
for every x in the domain of g,
and for every x in the domain of f.
  
f g x x

  
g f x x


14. Inverse Functions (cont.)
 Example: Decide whether g is the inverse function of f .
  3
1
f x x
    3
1
g x x
 

15. Inverse Functions (cont.)
 Example: Decide whether g is the inverse function of f .
yes
  3
1
f x x
    3
1
g x x
 
 
   
3
3
1 1
f g x x
  
1 1
x
  
x

 
  3 3
1 1
g f x x
  
3 3
x

x


16. Inverse Functions (cont.)
 If g is the inverse of a function f , then g is written as f -1
(read “f inverse”).
 In our previous example, for ,
  3
1
f x x
 
 
1 3
1
f x x

 

17. Finding Inverses
 Since the domain of f is the range of f -1 and vice versa, if
a set is one-to-one, then to find the inverse, we simply
exchange the independent and dependent variables.
 Example: If the relation is one-to-one, find the inverse of
the function.
         
 
2,1 , 1,0 , 0,1 , 1,2 , 2,2
F    not 1-1
       
 
3,1 , 0,2 , 2,3 , 4,0
G  1-1
       
 
1
1,3 , 2,0 , 3,2 , 0,4
G


18. Finding Inverses (cont.)
 In the same way we did the example, we can find the
inverse of a function by interchanging the x and y
variables.
 To find the equation of the inverse of y = f x:
 Determine whether the function is one-to-one.
 Replace f x with y if necessary.
 Switch x and y.
 Solve for y.
 Replace y with f -1x.

19. Finding Inverses (cont.)
 Example: Decide whether each equation defines a one-
to-one function. If so, find the equation of the inverse.
a)   2 5
f x x
 

20. Finding Inverses (cont.)
 Example: Decide whether each equation defines a one-
to-one function. If so, find the equation of the inverse.
a) one-to-one
replace f x with y
interchange x and y
solve for y
replace y with f -1x
  2 5
f x x
 
2 5
y x
 
2 5
x y
 
2 5
y x
 
5
2
x
y


 
1 1 5
2 2
f x x

 

21. Graphing Inverses
 Back in Geometry, when we studied reflections, it turned
out that the pattern for reflecting a figure across the line
y = x was to swap the x- and y-values.
 It turns out, if we were to graph our inverse functions,
we would see that the inverse is the reflection of the
original function across the line y = x.
 This can give you a way to check your work.
   
, ,
x y y x


22. Graphing Inverses
  2 5
f x x
     
3
2
f x x
 
 
1 1 5
2 2
f x x

   
1 3
2
f x x

 

23. Classwork
 4.1 Assignment (College Algebra)
 Page 413: 42-50; page 384: 22-30; page 376: 112-116
 4.1 Classwork Check
 Quiz 3.6