This document covers the concepts and objectives related to inverse, exponential, and logarithmic functions, including identifying one-to-one functions and finding their inverses. It discusses the composition of functions, provides examples of one-to-one functions, and explains how to determine inverse functions. Additionally, it includes methods for graphing inverses and illustrates the inverse relationship through reflection across the line y = x.

1. 4.1 Inverse Functions
Unit 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 .
a) Find
b) Find
  2 1f x x  

4
1
g x
x
  2f g
  3g f

5. Composition of Functions (cont.)
 Example: Let and .
a) Find
b) Find
  2 1f x x  

4
1
g x
x
  2f g
  3g f
  

4
2 4
2 1
g         2 2 4f g f g f
   2 4 1 7
        3 2 3 1 7f
     
  
4 4 1
7
7 1 8 2
g

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

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

8. 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.

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

10. 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 7f x x
   f a f b
3 7 3 7a b    
3 3a b  
a b

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

12. 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
2f x x 
  2
3 3 2 11f   
   
2
3 3 2 11f     

13. 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

14. 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 5f x x   
5
8
x
g x


   8 10 810 5 5f   
 
85 5 80
8
8
8
105g

  

15. 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

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

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

18. 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
1f x x 
 1 3
1f x x
 

19. 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 , 2,2F    not 1-1
        3,1 , 0,2 , 2,3 , 4,0G  1-1
        1
1,3 , 2,0 , 3,2 , 0,4G


20. 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.

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

22. 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 5f x x 
2 5y x 
2 5x y 
2 5y x 
5
2
x
y


 1 1 5
2 2
f x x
 

23. 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

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

25. Classwork
 College Algebra
 Page 413: 42-50, page 384: 22-30, page 352: 58-68