5.3 Inverse Function

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.

Definition of a Real-Valued Function of a Real Variable

Review

Function is a mapping!

ReviewRelation – 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.

Both relation and function are a mapping. A function is a relation but not vise versa.

Function Relation

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

Example 1 Tell the following relations are function or not:

1. {(1, 0), (-2, 3), (3, -1), (1, -2), (3, 0)}

2. {(2, 0), (-5, 0), (3, 0)}

3. {(0, -1), (-2, 3), (4, -1), (1, -2), (-6, -2)}

4. {(1, 0), (3, 2)}

5. {(1, 0), (0, 3), (3, -1), (1, 1), (6, -2), (9, 0)}

Not a function

Function

Function

Function

Not a function

What is the characteristic of the non-functions?At least two points on the same vertical line!

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.

Definition of Inverse Function and Figure 5.10

x1

y2

x2

y1

f –1 f

X

f (X)

What kind of function has inverse function?

Revisit Example 1Example 1 Tell the following relations are function or not: 2. {(2, 0), (-5, 0), (3, 0)}

3. {(0, -1), (-2, 3), (4, -1), (1, -2), (-6, -2)}

4. {(1, 0), (3, 2)}

Function

Function

Function

x1

y2

x2

y1

f –1 f

No inverse

No inverse

Has inverse

X

f (X)

What kind of function has inverse function?

Answer: A function has inverse must have the following properties:

1) For any two different x-values, their images must be different: If x1 ≠ x2 , then f(x1) ≠ f(x2).

2) 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.

How do we know a function is a one-to-one function by

graph?

Only one-to-one function has inverse function.

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.

y = 7

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).

(4, 7)

x

y

2

2

(0, 7)

one-to-one

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

not one-to-one

x

y

-4 4

4

8

x

y

-4 4

4

8

Theorem 5.7 The Existence of an Inverse Function

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.

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

y = f(x)y = x

y = f -1(x)

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.

The inverse function exists.

Reflect the graph of f in the line y = x to produce the graph of f -1.

x

y

Guidelines for Finding an Inverse Function

Note We should add 1a. into to the guideline1a. Find the domain and range of the f

Example 5 Find the inverse of the following function

Verify two functions are inverse each other algebraically

5 123)( xxfSolution

5 123 xy

5 123

xy

123

5

xy

1

32

15

yx

1

32

15

xy

Graphical test show this function has inverse Domain and range of this function are R, R.

1

32

1)(

51 x

xf

Domain and range of the inverse function are R, R.

Continued…

Verify the inverse function algebraically.

xxx

xxxff

33

33

113

31132

123))((

5

5

5

5

5

51

1

32

1)(

51 x

xf5 123)( xxf

xxx

xff

112

2

11

3

123

2

1))((

55

1

Example 6 Find the inverse of the following function

Find the inverse algebraically

2)(

xx eexf

Solution

2

xx eey

x

x

e

ey

2

12

12 2 xx eye

Graphical test show this function has inverse Domain and range of this function are R, R.

Note that and

0122 xx yee

12 yyex

yyyy 22 1

0xe

So,12 yyex

1lnln 2 yyex

1ln 2 yyx

Domain and range of the inverse function are R, R.

1ln 2 xxy

1ln)( 21 xxxf

Continued… 1ln 2 yyx

Verify the inverse function algebraically.

)1(2

1)1(

2

1))((

2

22

1ln

1ln2

12

2

xx

xx

e

exff

xx

xx

xxx

xxx

xx

xxxx

)1(2

)1(2

)1(2

11212

2

2

222

2)(

xx eexf

1ln)( 21 xxxfContinued…

1

22ln))((

2

1xxxx eeee

xff

2)(

xx eexf

14

2

2ln

22 xxxx eeee

4

2

2ln

22 xxxx eeee

2

22ln

xxxx eeee

22ln

xxxx eeee

xex ln

Theorem 5.8 Continuity and Differentiability of Inverse Functions

We now discuss the relationship between the derivative of the function and its inverse.Suppose that a function and its non-zero derivative are

)(xfy

dy

dxyf )()'( 1

And then its inverse is

xxffyf ))(()( 11

Taking the derivative of the inverse with respect of variable y, we have

)('

11

xfdxdy

))(('

11 yff

))(('

1)()'(

11

yffyf

or

0)(' xfdx

dy

0))((' 1 yffdx

dy

Since the variable y in this expression is only the dummy variable, so we change y to x.

))(('

1)()'(

11

xffxf

))(('

1)()'(

11

yffyf

The above is not a formal proof.

Example 7 Find the derivative of function

Application of Derivative of Inverse Function

xexg )(

Solution

0 ,ln)( xxxf

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

xexg

xgxgf

xg )(

)(11

))(('

1)('

0 ,01

)(' xx

xf By using the concept in the above example, we can find the derivatives for many functions.

Example 8 Let

Application of Derivative of Inverse Function

1)( 34

1 xxxf

SolutionNotice that f is one-to-one function and therefore has its inverse f -1.a. . So

3)2( f

a. What is the value of when )(1 xf 3x

b. What is the value of when )()'( 1 xf 3x

2)3(1 f

b. By the Theorem 5.9, we know

4

1

12

1

)2('

1

))3(('

1)3()'(

211

43

ffff

Example 9 Let (for x ≥ 0) and let

Application of Derivative of Inverse Function2)( xxf

Solution

The derivatives of f and f -1 are:

and

xxf 2)('

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)

xxf

2

1)()'( 1

At point (a, a2), the slope of graph of f is

xxf )(1

aaf 2)('

At point (a2, a), the slope of graph of f -1 is

aaaf

2

1

2

1)()'(

2

21

Homework

Pg. 347 23-29 odd, 33, 35, 71-75 odd