Download Standard Functions

College of Engineering
MATHEMATICS I
Common / Standard Functions
Dr Fuad M. Shareef
1
Relations / Mappings
Let A and B be two sets.
A binary relation, R, from A to B assigns to each
ordered pair (a,b) exactly one of the following
statements:
(i) “a is related to b”, written as aRb or (a,b)  R,
(ii) “a is not related to b", written as a b or (a,b)R.
Binary relation can be thought of as a statement which,
given any two elements of a set A ( an ordered pair ),
is either true or false for that pair.
We know the relation completely if we know the set of
pairs for which it is true.
2
Relations/ Mappings
Example:
Let A ={1,2,3}, and R={(1,2) ,(1,3),(3,2)}, then R is a
relation on A since it is a subset of AxA.
With respect to this relation:
1R2 , 1R3 , 3R2
But the following are not
1 R 1 , 2 R 1 , 2 R 2 , 2 R 3 , 3 R 1 , 3 R 3.
The domain of R is {1,3}
The range of R is {2,3}.
3
Domain and Range
Domain of a function:
• consists of all values which the rule may be
applied.
• The set of values that are valid in the rule
• The first coordinates of the ordered pairs
When we write a function, both the rule and the
domain is given.
If the domain is not given, then it is assumed to be
defined for all values that are valid in the rule.
4
Domain of functions
Given
Find:
f(2)
f(3)
f(0)
f ( x)  x  2
f (2)  2  2  0
f (3)  3  2  1
f (0)  0  2  2
the least possible and the largest
2 and 
possible domain for f
Domain is :
D  {x : x  2 , x  }
undefined
5
Domain of functions
Given
Find:
f(2)
f ( x) 
1
x3
1
1

23 5
1
1
f (3) 

undefined
f(-3)
3  3 0
1
1
f (0) 

f(0)
03 3
1
1
f (5) 

5  3
2
f(-5)
the least possible and the largest - and 
possible domain for f
Domain is :
D  {x : x  3 , x  }
f (2) 
6
Range of functions
Range of a function:
• The set of values produced
from applying the rule to
the domain of the
function.
• The set of out put values
of the function
Example1:
For
1
f ( x) 
x 1
,
x  1.
Example2:
For
f ( x) 
1
,
 x 1
x  1.
The range R is:
R  { y : y  0, y  }
The range R is:
R  { y : y  0, y  }
7
Graph of common functions
• Linear function
f ( x)  ax  b,
f(x)= -2x-3
a, b  , D  {x : x  }, R  { y : y  }
y
f(x)=2x+3
x
8
Graph of common functions
• Quadratic function- Parabola
f1 ( x)  x ,
f 2 ( x)   x2 ,
2
D  {x : x  }, R  { y : y  0, y  }
D  {x : x  }, R  { y : y  0, y  }
y
f1=x2
x
f2= - x2
9
Graph of common functions
• Cubic function
y  f ( x)  x3 ,
D  {x : x  }, R  { y : y  }
y
f ( x)  x
3
x
10
Graph of common functions
• Surd function –square, cubic, fourth (roots)
f ( x)  x ,
D  {x : x  0, x  }, R  { y : y  0, y  }
y
f ( x)  4 x
f ( x)  x
x
f ( x)  3 x
11
Exponential & Logarithmic
Functions
x
Exponential Function: f : x
a
Where x  and a  
f is called an exponential function.
y
Y=f(x)=(0.5)x
f(x)=4x
Y=3x Y=f(x)=2x
x
12
Exponential & Logarithmic
Functions
Logarithmic Function: f1 : x ln x, x  0
Or
f 2 : x log x, x  0
f1 is the natural logarithm
f 2 is the common logarithm
y
f1 : x
f2 : x
ln x
log x
x
13
Rational Functions
Rational Function:
f ( x)
ax n  ...
R( x) 
 m
,
g ( x)  bx  ...
where g ( x)  0
2
R1 ( x)  , x  0
x
x 3
More complicated: R2 ( x)  3
, x  0,1
2
x  2x  x
Simple example:
y
x 3
R2 ( x)  3
x  2x2  x
2
R1 ( x) 
x
x
?
The lines x=0 and x=1 are called Asymptotes.
14
Complete the graph of Exercise:
x 3
R2 ( x)  3
, x  0,1
2
x  2x  x
in the interval (0,1)
y
Notice when x  3, R2  0
The lines x=0 and x=1 are called Asymptotes
15
x
Trigonometric Functions
Sine Function: f ( x)  sin x,
x
Range:
R= {x : 1  x  1, x  }
y
y  f ( x)  sin x
x
16
Trigonometric Functions
Cosine Function: f ( x)  cos x,
x
Range:
R= {x : 1  x  1, x  }
y
y  f ( x)  cos x
x
17
Trigonometric Functions
n
Tangent Function: f ( x)  tan x, x   , n  1,3,5, 7...
2
Range:
R={y : y  }
y
y  f ( x)  tan x
x
18
Tutorial Exercise, Assignment and
Lecture notes
Visit:
the courses website once everyday.
19