1. CHAPTER 5 SECTION 5.3 INVERSE FUNCTIONS

2. Definition of Inverse Function and Figure 5.10

3. 1. Show that these functions are inverses:

4. 1. Show that these functions are inverses:
You could…
Graph each and show symmetry about y = x.
Show that both f(g(x)) = x and g(f(x)) = x.
Find the inverse of one of the functions and compare.
To find an inverse:
Swap x & y.
Solve for y.
Domain of f -1(x) = Range of f(x).
These functions are inverses.

5. Theorem 5.6 Reflective Property of Inverse Functions

6. Theorem 5.7 The Existence of an Inverse Function and Figure 5.13
Do you remember what monotonic means!!!!!!!!!!??????????????
PASSES HOROZONTAL LINE TEST!!

7. Inverse Functions
If f(g(x)) = x and g(f(x)) = x then f(x) and g(x) are inverses.
Domain of f(x) = Range of f -1(x)
Range of f(x) = Domain of f -1(x)
Inverses are symmetric about y = x.
A function can only have an inverse if it is 1-to-1.
2 ways to check 1-to-1: a. b.

8. Inverse Functions
If f(g(x)) = x and g(f(x)) = x then f(x) and g(x) are inverses.
Domain of f(x) = Range of f -1(x)
Range of f(x) = Domain of f -1(x)
Inverses are symmetric about y = x.
A function can only have an inverse if it is 1-to-1.
2 ways to check 1-to-1:
a. horizontal line test
b. is it always inc or dec?
Note: if a function isn’t 1-to-1 we can change its domain to make it 1-to-1.

9. Can these functions have inverses?

10. Can these functions have inverses?
This function can’t have an inverse.
This derivative is always positive, so y is always increasing.
This derivative changes signs, so y increases and decreases.
This function can have an inverse.
This function can’t have an inverse.
…unless we limit its domain to all positives or all negatives.

11. Guidelines for Finding an Inverse Function

12. 5. Find the inverse of

13. 5. Find the inverse of
This function isn’t 1-to-1.
Limit its domain.

14. Theorem 5.8 Continuity and Differentiability of Inverse Functions

15. Theorem 5.9 The Derivative of an Inverse Function

16. Example
What is the value of f-1 (x) when x = 3?
Since we want the inverse, 3 would be the y coordinate of some value of x in f(x).
As you can see, we could
try to guess an answer but
we have no means to solve
the equation. Let’s look at
the graph.

17. On the graph you can see that a y value of 3
corresponds to an x value of 2, thus if (2,3) is on the
f function, (3,2) is on the function.
f-1
So, f-1 (3) = 2
(2,3)

18. B. What is the value of (f-1)’ (x) when x = 3?
Solution: Since g’ (x) = 1/ f’ (g(x)) by Th 5.9, we can substitute f-1 for g, thus
[f-1 (x)]’ = 1/ f’ (f-1 (x))
[f-1 (3)]’ = 1/ f’ (f-1 (3)) = 1/ f’(2)
=1/(3/4(2)2+1) = 1/4

19. We showed previously that these functions are inverses.

20. We showed previously that these functions are inverses.
Conclusion:
If (a,b) is on f(x), that means (b,a) is on f -1(x), AND…

22. To find
a. Set b = f(a) .
Solve for a.

23. Graphs of Inverse Functions Have Reciprocal Slopes
Two inverse functions are:
Pick a point that satisfies
f, such as (3,9), then
(9,3) satisfies g.

24. Graphically: f and its inverse should look like mirror
Homework Examples:
4. Show that f and g are inverse functions (a) algebraically and (b) graphically
Solution: One way to do (a) is to show that f(g(x))=x and g(f(x)) = x. A second method would be to find the inverse of f and show that it is g.
Four steps to finding an Inverse:
Step 1 change f(x) to y
Step 2 Interchange x and y
Step 3 solve for y
Step 4 change y to f-1
Graphically: f and its inverse should look like mirror
images across the line y = x.

25. Graphically: f and its inverse should look like mirror
Homework Examples:
4. Show that f and g are inverse functions (a) algebraically and (b) graphically
Solution: One way to do (a) is to show that f(g(x))=x and g(f(x)) = x. A second method would be to find the inverse of f and show that it is g.
Four steps to finding an Inverse:
Step 1 change f(x) to y
Step 2 Interchange x and y
Step 3 solve for y
Step 4 change y to f-1
Graphically: f and its inverse should look like mirror
images across the line y = x.

26. Graphically: f and its inverse should look like mirror
Homework Examples:
4. Show that f and g are inverse functions (a) algebraically and (b) graphically
Solution: One way to do (a) is to show that f(g(x))=x and g(f(x)) = x. A second method would be to find the inverse of f and show that it is g.
Four steps to finding an Inverse:
Step 1 change f(x) to y
Step 2 Interchange x and y
Step 3 solve for y
Step 4 change y to f-1
Graphically: f and its inverse should look like mirror
images across the line y = x.

27. Graphically: f and its inverse should look like mirror
Homework Examples:
4. Show that f and g are inverse functions (a) algebraically and (b) graphically
Solution: One way to do (a) is to show that f(g(x))=x and g(f(x)) = x. A second method would be to find the inverse of f and show that it is g.
Four steps to finding an Inverse:
Step 1 change f(x) to y
Step 2 Interchange x and y
Step 3 solve for y
Step 4 change y to f-1
Graphically: f and its inverse should look like mirror
images across the line y = x.

28. Graphically: f and its inverse should look like mirror
Homework Examples:
4. Show that f and g are inverse functions (a) algebraically and (b) graphically
Solution: One way to do (a) is to show that f(g(x))=x and g(f(x)) = x. A second method would be to find the inverse of f and show that it is g.
Four steps to finding an Inverse:
Step 1 change f(x) to y
Step 2 Interchange x and y
Step 3 solve for y
Step 4 change y to f-1
Graphically: f and its inverse should look like mirror
images across the line y = x.

29. Graphically: f and its inverse should look like mirror
Homework Examples:
4. Show that f and g are inverse functions (a) algebraically and (b) graphically
Solution: One way to do (a) is to show that f(g(x))=x and g(f(x)) = x. A second method would be to find the inverse of f and show that it is g.
Four steps to finding an Inverse:
Step 1 change f(x) to y
Step 2 Interchange x and y
Step 3 solve for y
Step 4 change y to f-1
Graphically: f and its inverse should look like mirror
images across the line y = x.

30. 55. Show that f is strictly monotonic on the indicated interval and therefore has an inverse on that interval. (Strictly monotonic means that f is always increasing on a given interval or f is always decreasing on a given interval ).
On (0, )
The derivative is always
negative on (0, ), therefore, f is decreasing and thus has an inverse on this interval.

31. 55. Show that f is strictly monotonic on the indicated interval and therefore has an inverse on that interval. (Strictly monotonic means that f is always increasing on a given interval or f is always decreasing on a given interval ).
On (0, )

32. 55. Show that f is strictly monotonic on the indicated interval and therefore has an inverse on that interval. (Strictly monotonic means that f is always increasing on a given interval or f is always decreasing on a given interval ).
On (0, )
The derivative is always
negative on (0, ), therefore, f is decreasing and thus has an inverse on this interval.

33. Some function f(x) and its derivative f '(x) are continuous and differentiable for all real numbers, and some of the values for the functions are given in the below table: Based on the information given, answer the following questions: (a) Evaluate (b) At what value c is the graph of discontinuous? You may not use a graphing calculator • Difficulty:

34. Remember that                        , allowing us to find the derivative of an inverse function given only the original function. However, you'll need to be able to compute     .
Remember, a function and its inverse differ in that the input for one is the output for the other. Since the above table tells us that f(1) = 3, we can be sure that
Plug what you know into the formula we began part (a) with and we get
                                   
The table tells us that f '(1) = –2, so     .             .

35. (b) The formula we used in part (a), written more generally, says that
Therefore, whenever the denominator
equals 0, the fraction will be undefined. So, the answer to our question is the equivalent to the solution of the equation
           :
To solve this, first decide at what value of x does f ' equal 0. The only answer we can be sure of is when x = 2. Since f '(2) = 0, we can then say, by substitution, that            . This means the exact same thing as f(2) = a. Basically, when it's all said and done, we're just looking for f(2), so the answer is –1. To check, try and evaluate             :

36. Obviously, that derivative doesn't exist.