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1
Lesson 10
Inverse Functions
10A • Inverse functions • Exponential Function and Logarithmic Function
Course I
2
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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3
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.
4
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 =
5
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?
6
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 =
7
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
8
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)
9
Lesson 10
Hyperbolic Functions
10B • Catenary • Hyperbolic Function • Why Do We Call Them “Hyperbolic” ?
Course I
10
Catenary
Catenary
Hyperbolic Function
xx
xxxxxx
eeeexeexeex −
−−−
+−
=+
=−
= tanh,2
cosh,2
sinh
Hanging chain (Catenary)
x
y
O
2xy =
12
−+
=−xx eey
Catenary
parabola
11
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
12
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
13
Example Example 2. Prove the following addition formula.
Ans.
yxyxyx sinhsinhcoshcosh)cosh( +=+
( ) ( )
( ) )cosh(21
41
41
2222sinhsinhcoshcosh
yxee
eeeeeeeeeeeeeeee
eeeeeeeeyxyx
yxyx
yxyxyxyxyxyxyxyx
yyxxyyxx
+=+=
+−−++++=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟⎠
⎞⎜⎜⎝
⎛ +⎟⎟⎠
⎞⎜⎜⎝
⎛ +=+
−−+
−−−−−−−−
−−−−
14
Exercise
Pause the video and solve the problem.
Ans.
Exercise 2. Prove the following formulae. (1) (2)
1sinhcosh 22 =− xx
yxyxyx sinhcoshcoshsinh)sinh( +=+
15
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
+=−=
−+−+−−+=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛ ++⎟⎟⎠
⎞⎜⎜⎝
⎛ +⎟⎟⎠
⎞⎜⎜⎝
⎛ −=+
−−+
−−−−−−−−
−−−−