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5.3 Inverse Function
Review for Algebra Two:
Definition of Inverse Function
A function g is the inverse function of the function f if f g x   x for each x in the
domain of g and g  f x   x for each x in the domain of f.
The function g is denoted by f 1 (read “f inverse”)
Example:
f x  x  3 , g x is its inverse function, which is g x   f 1 x   x  3
Find the inverse functions. Verify them by the definition of inverse function
I do:
f x  3x  2
We do:
f x   x 3
You do:
f x   2 x 3  1
We all do:
f x   2 x  3
Use TI 84 to graph an inverse function
http://www.tc3.edu/instruct/sbrown/ti83/drawinv.htm
Continuity & Differentiability of Inverse Functions
Let f be a function whose domain is an interval I. If f has an inverse function, then the
following statement are true..
1. If f is continuous on its domain, then f 1 is continuous on its domain.
2. If f is increasing on its domain, then f 1 is increasing on its domain.
3. If f is decreasing its domain, then f 1 is decreasing on its domain.
4. If f is differentiable on an interval containing c and f ' c   0 , then f 1 is
differentiable at f (c).
The derivative of an Inverse Function
Let f be a function that is differentiable on an interval I. If f has an inverse function
g, then g is differentiable at any x for which f ' g x   0 .
1
Moreover, g ' x  
, f ' g x   0
f ' g x 
I do:
Let f  x  
1 3
x  x 1
4
a) What is the value of f
1

b) What is the value of f
x  when x = 3?
' x  when x = 3?
1
We do:
Let f x   x 3  1
a) What is the value of f 1 x  when x = 26?
b) What is the value of f 1 ' x  when x =26?
 
Let f x   5  2 x 3
c) What is the value of f 1 x  when x = 7?
d) What is the value of f 1 ' x  when x = 7?
 
Homework:
Text P. 350 #73-to 79 odd
5.6 Inverse Trigonometric Functions: Differentiation
Definitions of Inverse Trigonometric Functions
Function
Domain
y = arcsin x iff sin y = x
1  x  1
y = arccos x iff cos y = x
y = arctan x iff tan y = x
1  x  1
  x  
y = arccot x iff cot y = x
y = arcsec x iff sec y = x
  x  
Range


  y
2
2
0 y 


  y
2
2
0 y 
x 1
0 y  , y 
y = arccsc x iff csc y = x
x 1
Evaluate each function.
I do:
 1
arcsin   
 2


2
 y

2
We do:
arccos 0
Use TI 84 to verify your answer.
Properties of Inverse Trigonometric Functions


If  1  x  1 and   y  , then sin arcsin x  x and sin arcsin y   y
2
2
Similar properties hold for the other inverse trigonometric functions.
arctan 2 x  3 

4
We do:
arcsin 3x    
1
2
2
, y0
You do:
arctan 3
I do:

You do:
arctan 2x  5  1
Derivatives of Inverse Trigonometric Functions
Let u be a differentiable function of x.
d
d
arcsin u  u' 2
arccos u   u' 2
dx
dx
1 u
1 u
d
d
arctan u   u ' 2
arc cot u    u '2
dx
dx
1 u
1 u
d
u'
d
 u'
arc sec u  

arc csc u  
dx
dx
u u 2 1
u u 2 1
I do:
d
arcsin 2 x  
dx
We do:
d
arctan 3x  
dx
You do:
d
arcsin
dx


x 
We all do:
d
arc sec e 2 x 
dx


We all do:
y  arctan x  x 1  x 2
Find y’
Homework:
Text P. 379 #5 to 11 odd #39, #43 to 53 odd
5.7 Inverse Trigonometric Functions: Integration
Since the derivatives of the six inverse trigonometric functions fall into three pairs, we
only need to remember one from each pair for the integration.
Let u be a differentiable function of x, and let a > 0,
1

dx  arcsin x  C
1 x2
1
 1  x 2 dx  arctan x  C
1
 x x 2  1dx  arc sec x  C

1
du  arcsin
a2  u2
1
1
u
 a 2  u 2 du  a arctan a  C
u
1
1
du

arc
sec
C
 u u2  a2
a
a
*Only need to remember the right hand columns.
I do:

1
4  x2
dx 
You do:
1
x
4x 2  9
You do:
1

e 2x  1
dx 
dx 
u
C
a
We do:
1
 2  9x 2 dx 
We all do:
x2
 4  x 2 dx 
Completing the Square
2
2
b
b

x  bx  c  x  bx        c   x 
2
2

2
2
2
I do:
x
2
1
dx 
 4x  7
We do:
Find the area of the region bounded by the graph of
1
f x  
3x  x 2
3
9
The x axis, the line x  , and the line x 
2
4
You do:
2
dx
0 x 2  2x  2
Review:
P. 386
When we have a integral with fraction
2
b
b
   c
2
2
Power Rule
Log Rule
Inverse
Trig rules
Can not find this integral using the techniques you have learned
Homework:
Text P. #3 to 13 odd #25 to #29 odd #41