5.3 Inverse Function. After this lesson, you should be able to: Verify that one function is the...
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Transcript of 5.3 Inverse Function. After this lesson, you should be able to: Verify that one function is the...
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