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Lesson １０

Inverse Functions

１０A •  Inverse functions •  Exponential Function and Logarithmic Function

Course I

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Inverse Function – Example 1

Example : Consider

The inverse function is given by exchanging and . yx

121

+= yx that is

1

1.5

1.5

1 1 1.5

)1(5.1 f=

)5.1(1 g=

The inverse function bring back to 1.

)(xg)1(f

121)( +== xxfy

22)( −== xxgy

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Inverse Function

Definition of Inverse Functions

Functions and are inverses of one another if: and for all values of x in their respective domains.

)(xf )(xg( ) ( ) xxfgxxgf == )()(

How to Find Inverse Function

1.  Replace the variables.

2.  Rearrange the expression.

3.  Determine that the domain of is the same as the range of .

)()( yfxxfy =→= 　

)()( xgyyfx =→= 　

)(xg)(xf

1

( )20

121

≤≤

+=

x

xy

( )2122

≤≤−=xxy

Domain of

Range of )(xf

)(xg

1

2

2

x

y

Symmetrical about x=y.

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Inverse Function – Example(2)

Parabola

Not inverse

Because (b) is not a function

(b) (a)

(c) (d)

Inverse function of each other

x

y

O

x

y

O

xy =

xy −=

x

y

O 1

1

y = x2 x ≥ 0( )

y = x

)0(2 xxy ≤=

xy =x

y

O 1

1

y = x√

y = x

xy =

xy =

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Exponential Function & Logarithmic Function Exponential function ( )0,0 ≠>= aaay x

•  The domain is the positive real number. •  The range is the whole real number. •  The graph always passes the point (1, 0)

xy alog=

x

y

O 1

1

1>a

xay =

xy alog=

x

y

O 1

1

10 << a

xay =

xy alog=

The inverse function of is called a logarithmic function and written as

Uhh…. What?

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Example Example 1. Illustrate the function and its inverse function on the given coordinate plane.

Ans.

xy −−= 1

xy −−= 1

xy

1 0 -1 -2 -3 0 1 2 2 3

• Take symmetrical points about x=y, that is, •  replace their values

xy 1 0 -1 -2 -3

0 1 2 2 3

−3 −2 −1 1 2 3 x

−3

−2

−1

1

2

3

y

O

xy −−= 1

2xy −=

xy =

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Exercise

Pause the video and solve the problem.

Ans.

Exercise 1. (1) Illustrate the function (2) Illustrate the inverse function by mapping symmetrically about (3) Find the domain of the inverse function. (4) Find the expression of the inverse function by replacing and .

)0(2

≥+

= xxxy

xy =

yx

−5 −4 −3 −2 −1 1 2 3 4 x

−5−4−3−2−1

1234y

O

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Answer to the Exercise

Ans.

Exercise 1. (1) Illustrate the function (2) Illustrate the inverse function by mapping symmetrically about (3) Find the domain and the range of the inverse function. (4) Find the expression of the inverse function by replacing and .

)0(2

≥+

= xxxy

xy =

yx

221+

−=x

y

(1)  This expression becomes

(3) From the figure Domain : Range:

10 <≤ x0≥y

12

2 −−

=∴+

=xxy

yyx

−5 −4 −3 −2 −1 1 2 3 4 x

−5−4−3−2−1

1234y

O

12−

−=xxy

2+=xxy

(2) . See the figure.

(4)

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Lesson １０

Hyperbolic Functions 　

10B • Catenary •  Hyperbolic Function •  Why Do We Call Them “Hyperbolic” ?

Course I

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Catenary

Catenary

Hyperbolic Function

xx

xxxxxx

eeeexeexeex −

−−−

+−

=+

=−

= tanh,2

cosh,2

sinh

Hanging chain (Catenary)

x

y

O

2xy =

12

−+

=−xx eey

Catenary

parabola

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Fundamental Characteristics of Hyperbolic Functions Trigonometric functions

xx

xxxxx

22

22

cos1tan1

1sincoscossintan

=+

=+

=

xx

xxxxx

22

22

cosh1tanh1

1sinhcoshcoshsinhtanh

=−

=−

=

Graphs

xy sinh=

xy cosh=

xy tanh= x

y

O

1

-1Hyperbolic functions

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Why Do We Call Them “Hyperbolic” ?

Comparison of Two Parametric Expressions )sin,(cos),( ttyx =

)sinh,(cosh),( ttyx =

Eliminating the parameter, we have 122 =+ yx

From the previous slide, we have

122 =− yxThis expression represents a hyperbola.

This expression represents a circle.

► Based on the similarity in shape, we call them hyperbolic functions. ► Based on the similarity in character, we use the symbols etc. xsinh

x

y

O

x

y

O

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Example Example 2. Prove the following addition formula.

Ans.

yxyxyx sinhsinhcoshcosh)cosh( +=+

( ) ( )

( ) )cosh(21

41

41

2222sinhsinhcoshcosh

yxee

eeeeeeeeeeeeeeee

eeeeeeeeyxyx

yxyx

yxyxyxyxyxyxyxyx

yyxxyyxx

+=+=

+−−++++=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛ +=+

−−+

−−−−−−−−

−−−−

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Exercise

Pause the video and solve the problem.

Ans.

Exercise 2. Prove the following formulae. (1) (2)

1sinhcosh 22 =− xx

yxyxyx sinhcoshcoshsinh)sinh( +=+

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Answer to the Exercise

Ans.

Exercise 2. Prove the following formulae. (1) (2)

1sinhcosh 22 =− xx

(1) (2)

( ) ( ) 12412

41

22sinhcosh

2222

2222

=+−−++=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=−

−−

−−

xxxx

xxxx

eeee

eeeexx

yxyxyx sinhcoshcoshsinh)sinh( +=+

( ) ( )

( ) )sinh(21

41

41

2222sinhcoshcoshsinh

yxee

eeeeeeeeeeeeeeee

eeeeeeeeyxyx

yxyx

yxyxyxyxyxyxyxyx

yyxxyyxx

+=−=

−+−+−−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛ −=+

−−+

−−−−−−−−

−−−−