Transcript Slide 1
Inverse of Transcendental
Functions
1- Inverse of Trigonometric Functions
2- Inverse of Exponential Functions
3- Inverse of Hyperbolic Functions
1- Inverse of Trigonometric Functions
Since the trigonometric functions are not one-to-one, so they
don’t have inverse functions. However, if we restrict their
domains, then we may obtain one-to-one functions that have
the same values as the trigonometric functions and that have
inverse over these restricted domains.
For example, the function y sin x is not one –to-one on
its natural domain R. However, when the domain is
restricted to the interval –π/2 to π/2, it becomes one-to-one.
y
Graph of y sin x
1
y
x
2 3 / 2 / 2
/2
1
1
x
/2
/ 2
1
1
y
y sin x
/2
x
1
1
/ 2
3 / 2
2
Important Rules
*
1
y sin x sin y x
1
* sin sin x x , if 1 x 1
* sin
1
sin x x ,
if
2
x
2
Example
Find the domain of
f x sin 1 x2 1
Solution
D : 1 x2 1 1
0 x2 2
D: 2 x 2
y
Graph of y cos x
1
y
2 3 / 2 / 2
/2
3 / 2
1
1
/2
x
1
1
y cos x
y
/2
x
1
1
2
x
Important Rules
*
1
y cos x cos y x
1
* cos cos x x , if 1 x 1
* cos
1
cos x x ,
if 0 x
y
Graph of y tan x
x
3 / 2
y
/2
/ 2
x
1
y tan x
y
/2
x
/ 2
/ 2
/2
3 / 2
Important Rules
*
1
y tan x tan y x
1
* tan tan x x , if x
* tan
1
tan x x ,
tan
1
2
if
2
x
2
tan
1
2
Example
1
1
lim tan
x2
x2
Evaluate
Solution
x 2
1
x2
1
lim tan
x 2
x2 2
1
Notes
sin x sin x
1
cos x cos x
1
tan x tan x
1
sin
1
1
1
sin x
1
csc x
sin x
1
1
sec x
cos x
1
1
cot x
tan x
cos x
tan x
1
cos1
1/ 2 / 6
tan
1
1
1 / 4
3/2
/6
Important Rules
*
*
*
1
1
1/ x
1
1
1/ x
1
1
1/ x
csc x sin
sec x cos
cot x tan
1
Proof
csc x sin
1
1/ x
y csc x
1
1
1
sin y
x csc y
x csc y
sin 1/ x sin sin y
1
1
y csc x sin 1/ x
1
1
Example
Evaluate the given inverse function
i ) sec
1
3
ii ) cot
1
2.474
Solution
1
i ) sec 3 cos 1.910633236
3
1
1
1
ii ) cot 2.474 tan
0.3840267299
2.474
1
1
2- Inverse Exponential Functions
x
Every exponential function of the form a is a one-to-one
function. It therefore has an inverse function, which is
called the logarithmic function with base a and is denoted
by loga x .
y
ax
loga x
1
1
Domain:
(0, )
x
Range:
R (, )
The Natural Logarithmic Function
The logarithm with base e is called the natural logarithm and
has a special notation
loge x ln x
y ln x
y e x
y
1
x
1
Domaim : (0, )
Rnge : R
Basic Properties of Natural Logarithmic Function
lne x
x
e
lnx y ln x ln y
ln x
x
lnx / y ln x ln y
r ln x
ln x
ln 0
r
ln
Example
Solve the following equations for x
a) e
5 3 x
10
ln e
ln 10
5 3x ln 10
1
x 5 ln10 0.8991
3
b ) ln x 1 5
Solution
53x
2
e
e5
2
ln x 1
x
2
1 e
5
x e 1
2
5
x e 1 12.141382.
5
Example
Sketch the function
f x ln x 2 1
Solution
y
y
x
y
x=2
x
x
3- Inverse Hyperbolic Functions
The hyperbolic functions sinh x is one-to-one functions
1
and so they have inverse functions denoted by sinh x
1) sinh 1 x ln x x 2 1 ,
1
x
2) cosh x ln x x 1 , x 1
1 x
3) tanh x ln
,
1 x
1
2
1 x 1
1
sinh x ln x x 1 ,
Proof (1)
2
e e
x sinh y x
2
y
y sinh x
1
x R
y
e e 2x e 2x e 0
y
e
y 2
y
2 x e 1 0
y
2x 4x 4
e
2
2
y
y
y
e x x 1
y
2
y ln x x 1
2
Proof (3)
tanh
y tanh x
1
e
y
e
y
1
e
1 x 1
x tanh y x
x e
y
e
e y 1 x e y 1 x
y
1 x
x ln
,
1 x
1 x
1 x
y
e
y
xe
y
y
y
y
e e
e e
y
e
y
xe
1 x 1 x
e
y
2
y ln
1 x
1 x
y
Important Rules
sech 1 x cosh 1 1 / x
1
1
csc h x sinh
coth
1
x tanh
1 / x
1
1 / x