Download Algebraic Functions

‫بسم هللا الرحمن الرحيم‬
‫سبحان هللا و بحمده سبحان هللا العظيم‬
‫‪• 2.2 Identifying functions‬‬
Linear function
f(x)=m x+ c ;
m is the slope
c is the y-intercept
A constant functions has slope m=0 ;f(x)=c
(0,c)
f(x)=y=c
(0,c)
Polynomial functions
P (x )  an x n  an 1 x n 1  an  2 x n  2  ...........  a1 x  a0
where n is Integer and n> 0
a , a , a , ..a , a
the coefficients of polynomial
The domain of any polynomial is Dp =Re
If an  0 the degree of the polynomial is n
n
n 1
n 2
1
0
Polynomial functions
Degre? Domain?
Degree 3
Degree 4
Degree5
Degree 6
Power Functions
a
a is constant
f (x )  x
Integers
Rational
Domain?
a>0
a <0
Power Functions
a is constant
a is Rational
Algebraic Functions
polynomial using algebraic operations ( ,  ,  , /)
a ) f (x )  (x  5) x 2  9  3x 3
b )g (x )  5 x 2 (2 x  7)
x  5x 3
2
c ) h (x ) 

x
(3

x
)
3
x x
Rational Functions
ratio of two polynomials
f (x ) 
Domain? q (x )  0
g (x ) 
x 3
x 2 1
p (x )
, q (x )  0
q (x )
Notice:
All rational functions are special cases of algebraic
functions
functions
Algebraic
Transcendental
Algebraic Functions
Linear
Polynomial
Power Functions
Rational
Transcendental Functions
Trigonometric
f (x )  sin(x ) , h (x )  cos(x )
Exponential
g (x )  a x
Logarithmic
f (x )  loga x
Trigonometric Functions
f (x )  sin(x ) , h (x )  cos(x )
Exponential
g (x )  a
x
, a0
Logarithmic functions
The domain is (0,∞),
the range is (-∞,∞)
the function increases
slowly when x>1
f (x )  loga x , a  1, a  0
Transcendental Functions
The set of transcendental Functions includes the
• Trigonometric
• Inverse trigonometric
• Exponential
• Logarithmic
• Many other functions as well
Definitions 2.2.1:Increasing function
Let y=f(x) defined on an interval  and x 1 , x 2 
Increasing on  if x 1  x 2  f (x 1 )  f (x 2 )
f (x )  x 2 , x  (0, )
Definitions 2.2.1:Decreasing function
Let y=f(x) defined on an interval  and x 1 , x 2 
Decreasing on  if x 1  x 2  f (x 1 )  f (x 2 )
f (x )  x 2 , x  (,0)
Even functions and Odd functions
Even
Odd
f (x )  f (x ) x  D f
f (x )  f (x ) x  D f
Even functions and Odd functions
Even
Odd
Graph is symmetric about y-axis
f (x )  f (x ) x  D f
Graph is symmetric about Origin
f (x )  f (x ) x  D f
The domain must be symmetric
Example 2
Example:
Recognize the even and odd function
a ) f (x )  x 4  x 2
b ) g (x )  x 5  x 3  3x
x 2 1
c ) h (x )  3
x 1
d ) f (x )  x 5  x 3  3x  7
‫• تعلمت أن المعرفة لم تعد قوة في عصر السرعة واإلنترنت‬
‫والكمبيوتر‪ ،‬إنما تطبيق المعرفة هو القوة ‪.‬‬
‫• تعلمت أن الذين لديهم الجرأة على مواجهة الفشل ‪ ،‬هم‬
‫الذين يقهرون الصعاب وينجحون ‪.‬‬
‫• تعلمت أن المتسلق الجيد يركز على هدفه وال ينظر إلى‬
‫األسفل ‪ ،‬حيث المخاطر التي تشتت الذهن‬