1. 5.3 Inverse Function

2. After this lesson, you should be able to:
Verify that one function is the inverse function of another function.
Determine whether a function has an inverse function.
Find the derivative of an inverse function.

3. Definition of a Real-Valued Function of a Real Variable
Review
Definition of a Real-Valued Function of a Real Variable
Function is a mapping!

4. Review Relation – a set of ordered pairs.
Function – a set of ordered pairs in which no two ordered pairs have the same x-value and different y-values.
A function is a rule or correspondence which associates to each number x in a set X a unique number f(x) in a set Y
Both relation and function are a mapping. A function is a relation but not vise versa.
Function
Relation

5. Example 1 Tell the following relations are function or not:
1. {(1, 0), (-2, 3), (3, -1), (1, -2), (3, 0)}
Not a function
2. {(2, 0), (-5, 0), (3, 0)}
Function
3. {(0, -1), (-2, 3), (4, -1), (1, -2), (-6, -2)}
Function
4. {(1, 0), (3, 2)}
Function
5. {(1, 0), (0, 3), (3, -1), (1, 1), (6, -2), (9, 0)}
Not a function
What is the characteristic of the non-functions?
At least two points on the same vertical line!

6. Vertical Line Test (a graphical test for a function) See P. 22
A graph is a function graph if and only if every vertical line intersects the graph at most ONCE.

7. Definition of Inverse Function and Figure 5.10X x2 x1 f
f –1 y2 y1
f (X)

8. What kind of function has inverse function? Revisit Example 1
Example 1 Tell the following relations are function or not:
2. {(2, 0), (-5, 0), (3, 0)}
Function
No inverse
3. {(0, -1), (-2, 3), (4, -1), (1, -2), (-6, -2)}
Function
No inverse
4. {(1, 0), (3, 2)}
Function
Has inverse X x2 x1 f
f –1 y2 y1
f (X)

9. What kind of function has inverse function?
Answer: A function has inverse must have the following properties:
For any two different x-values, their images must be different: If x1 ≠ x2 , then f(x1) ≠ f(x2).
For any y-value in f(X), there exists an original image in X: For any y ∈ f(X), there exists x ∈ X such that
y = f(x).
This kind of function is called one-to-one function.
Only one-to-one function has inverse function.
How do we know a function is a one-to-one function by
graph?

10. Horizontal Line Test (a graphical test for a one-to-one function) See P. 343
A function has an inverse if and only if every horizontal line intersects the graph of a function at most ONCE.

11. Example 2 The function y = x2 – 4x + 7 is not one-to-one on the real numbers because the line y = 7 intersects the graph at both (0, 7) and (4, 7). x y 2
(0, 7)
y = 7
(4, 7)

12. Example: Horizontal Line Test
Example 3 Apply the horizontal line test to the graphs below to determine if the functions are one-to-one.
a) y = x3
b) y = x3 + 3x2 – x – 1 x y -4 4 8 x y -4 4 8
one-to-one
not one-to-one
Example: Horizontal Line Test

13. Theorem 5.7 The Existence of an Inverse Function

14. Theorem 5.6 Reflective Property of Inverse Functions and Figure 5.12
The domain of the inverse relation is the range of the original function.
The range of the inverse relation is the domain of the original function.

15. What is the relationship between the graph of the function and the graph of its inverse function?
Their graphs are symmetry to the line y = x

16. Example: Determine Inverse Function
Example 4 From the graph of the function y = f (x), determine if the inverse function exists and, if it does, sketch the graph of inverse.
The graph of f passes the horizontal line test. y
y = f -1(x)
y = x
y = f(x)
The inverse function exists. x
Reflect the graph of f in the line y = x to produce the graph of f -1.
Example: Determine Inverse Function

17. Guidelines for Finding an Inverse Function
Note We should add 1a. into to the guideline
1a. Find the domain and range of the f

18. Verify two functions are inverse each other algebraically
Example 5 Find the inverse of the following function
Solution
Graphical test show this function has inverse
Domain and range of this function are R, R.
Domain and range of the inverse function are R, R.

19. Continued…
Verify the inverse function algebraically.

20. Find the inverse algebraically
Example 6 Find the inverse of the following function
Solution
Graphical test show this function has inverse
Domain and range of this function are R, R.
Note that and
So,

21. Continued…
Domain and range of the inverse function are R, R.
Verify the inverse function algebraically.

22. Continued…

23. Theorem 5.8 Continuity and Differentiability of Inverse Functions

24. We now discuss the relationship between the derivative of the function and its inverse.
Suppose that a function and its non-zero derivative are
And then its inverse is
Taking the derivative of the inverse with respect of variable y, we have or

25. Since the variable y in this expression is only the dummy variable, so we change y to x.
The above is not a formal proof.

26. Application of Derivative of Inverse Function
Example 7 Find the derivative of function
Solution
It is kind of hard to find the derivative of g directly. Let’s consider another function
It is easy to know that g and f are inverse each other. And we know the derivative of f in the previous section. So
By using the concept in the above example, we can find the derivatives for many functions.

27. Application of Derivative of Inverse Function
Example 8 Let
a. What is the value of when
b. What is the value of when
Solution
Notice that f is one-to-one function and therefore has its inverse f -1.
a So
b. By the Theorem 5.9, we know

28. Application of Derivative of Inverse Function
Example 9 Let (for x ≥ 0) and let
Show that the slopes of graphs of f and f -1 are reciprocal at each of the following points:
(a, a2) and (a2, a) (a>0)
Solution
The derivatives of f and f -1 are:
and
At point (a, a2), the slope of graph of f is
At point (a2, a), the slope of graph of f -1 is

29. Homework
Pg odd, 33, 35, odd