a function relate an input to output and a set of output with ......each element x of x, at most one...
TRANSCRIPT
Functions: Definition
• A function relate an input to output
In mathematics, a function is a relation between a set of outputs
and a set of output with the property that each input is related
to exactly one output.
• of all the x for A function f: X to Y is a rule that assigns, to
each element x of X, at most one element of Y. If an
element is assigned to x in X, it is donated by f(x). The
subset of X consisting which f(x) is defined is called the
Domain of f .The set of all element in Y of the form of f(x),
is called the range of f
• General form for a function in many independence
variables
Y=G(Xi), i = 1,2,3,…,n
• Where :Y: is dependent variable, X: are independent
variables.
An example
•
• Defines y as a function of x. The equation gives the
rule add 2 to the value of x
• Which means when x=2, y=4 and when x = -6 then
become y=-4
• F(x), which is read f of x and which means the
output, in the range of f , that results when the rule f
is applied to the input x , from the domain of f.
• The outputs is called Function values .
Equality of Functions:
To say that two functions f and g in terms of x and y
are equal, denoted g=f ,is to say that :
• The domain of f is equal to the domain of g .
• For every x in the domain of f and g .f(x)=g(x).
Example : Determine whether the following
functions are equal .
f(x)=x+2
g(x)=(x+2)(x-1)∕(x-1)
It is easy to find that f and g are equal…
Exercise(1)
Determine whether the given functions are equal.
1. F(x)=x , g(x)= √ .
2. H(x)=x+1 , f(x)=(√ )2.
Find the indicated value for the given function:
1. f(x)=2x+1 , f(0) , f(3) ,f(-4).
2. f(u)=2u2-u ,f(-2) , f(2v) ,f(x+a).
3. h(x)=(x+4)2 ,h(0) , h(2) ,h(t-4) .
Special functions :
• Term special concerns about functions having special
forms and representations, and we will begin with
the simplest type, which is called constant function .
Constant function
• we call h a constant function because all the
function values are the same , in a more specific way
it is a function of the form:
• h(x) = c ,
• where c is a constant , is called a constant function .
polynomial function
• A constant function belongs to a boarder class of
functions, called polynomial function .
• In general , a function of the form
f(x)=qxn+wxn-1+exn-2+……..+c .
• n is a positive integer ,where q, w, e and c are
constants.
• It is called polynomial function , n is the degree of a
polynomial
• And q is the leading coefficient .
• polynomial function can be function in one
dependent variable or more than one variable.
• The Linear function
• is a function of degree one, take the form:
•
• The square polynomial function :
• Is polynomial function of the degree 2.
• Cubic function
• Is Polynomial function of degree 3.
It is take the form :
nixaxfy
xaxaxaaxfy
i
i
a
,..,3,2,1)(
)( 3
3
2
20
ni،xaxfy
xaxaxaxaaxf
i
iii
n
nni
,....,3,2,1
......)( 3
33
2
22110
Examples of a polynomial functions
•
Is a polynomial function of degree 3 with leading
coefficient 1
•
Is a linear function with leading coefficient 2∕3
• Rational Functions
• A rational function is formed by dividing one
polynomial by another polynomial ,
• f(x)=
,
• for example:
•
• Note that the polynomial function is a rational
function but the denominator equals 1
• Example :
•
.
Exercise (2)
1/ Determine whether the given functions is a
polynomial function :
• f(x) = X2 – x4 +4
• g(x)=7∕x+4
• j(x) = 2-3x
2/ Determine whether the given functions is a rational
function :
• f(x) =(x2+x)∕(x3+4)
• g(x)= 4x-4
• m(x)= 3∕2x+1
3/ state the degree and the leading coefficient of the
given polynomial function :
• g(x)= 9x2 +2x +1
• f(x)=1∕π -3x5 +2x6 + x7
• e(x)= 9
4/ find the function values for each function:
• g(x) = 3- x2 , g(10) , g(3) , g(-3)
• f(x)=8 , f(2) , f(t+8)
Combinations of Functions :
There are many ways of combining two functions to
create a new one . suppose f and g are the functions
given by :
f(x)=7x g(x)=2x2
• by adding : h(x)=7x+2x2
• by subtracting :
r(x)= 7x-2x2
or :e(x)= 2x2-7x
• by multiplying: t(x)=14x3.
• By dividing: c(x)=0.3x
Note : for each new function the resultant domain is
set of all x which belong to both the domain of f and g
• Also : (cf)(x)= c.f(x)
Exercise (3)
1. If f(x)= 3x-1 , g(x)=x2+ 3x , find :
2. (f+g)(x)
3. (f-g)(x)
4. (fg)(x)
5. (f∕g) (x)
6. (0.5f)(x)
Exponential Functions :
The functions of the form f(x) = bx , for constant b , are
important in mathematics , business , economics ,
science and other areas of study . An example is f(x) =
2x . such functions are called exponential functions .
Definition
The function f defined by :
F(x) = bx
Where b ≥ 0 , b ≠1 , and the exponent x is any real
number , is called an Exponential Functions with base
b .
Rules for exponents :
bx by = bx+y
(bx)y = bxy
(b c)x = bx cx
(
)
b1 = b
b0 = 1
(b . c-1)x = bx c-x
Example 1 : Bacteria Growth:
The number of bacteria present in a culture after t
minutes is given by :
N(t) = 300 (4∕3)t
How many bacteria are present initially?
Solution:
Here we want to find N(t) when t =0. we have:
(
)
Thus , 300 bacteria are initially present .
Approximately how many bacteria are present after 4
minits?
Exponential Function with base e :
The number e provides the most important base for
an exponential function. In fact the Exponential
function with base e is called the natural exponential
function and even the exponential function to stress it
is importance .
It has a remarkable property in calculus . it also occurs
in economic analysis and problem involving
exponential growth,
Y= ex
Logarithmic Functions
Y= logb x if and only by=x
And we have:
log b bx= x ………………(1)
blogb
x = x …………….….(2)
Where equation (1) holds for all x in (- ∞, + ∞) the
domain of the exponential function with base b
And equation (2) holds for all x in (0 , ∞) the range of
the exponential function with base b
Logarithmic Function properties
1- Logb (mn) = logb m + logb n .
And logb m + logb n = logb (mn).
For example :
Log 56 = log (8 . 7) = log 8 + log 7
2- Logb (m∕n) = logb m – logb n
For example :
Log (9∕2) = log 9 – log 2
3- Logb mr = r logb m
Example :
log 64 = log 82 = 2 log 23
= 2 * 3 log 2 = 6 log 2 .
4- Log (1∕m) = - log m
Example :
Log ¼ = -log 4
log(2∕3) = - log (3∕2)
Piecewise –Defined Function
• Let
• F(x)= -2 x if x≤ -3
3x-1 If -3≤ x ≤ 2
-4x if x ≥2
• This is called Piecewise –Defined Function,
because the rule for specifying it is given by rules
for each of several disjoin case.
• Where s is the independent variable, and the
domain of F is all (S) such that
• F(-5)= -2 (-5) = 10 (-5, 10)
• F (-1) = 3 (-1) -1= -4 (-1,-4)
• F (-3) = 3(-3) -1= -10 (-3,-10)
• F (4)= -4x =-4 (4)= -16 (4,-16)
Piecewise –Defined Function:
• e.g
• F(x) = x-2 if x <3
5-x if x ≥ 3
• F (-5)= x-2= -5-2=07
• F (-1)= x-2= -1-2=-3
• F (0)= x-2= 0-2=-2
• F (3)= 5-x= 5-3=2
• F (5)= 5-5=0
Absolute –value function:
The function | -1|(x)= | 1| is called absolute value
function.
Recall that absolute value of real number x is a
function denoted by
| x| and defined by
| x|= x if x ≥0
-x if x ≤0
Thus the domain of |- | is the real numbers. Some
functions value are
| 16|= 16
| -4/3|= - | -4/3|= 4/3
| 0|= | 0|=
e.g
f (x)= | 2x + 6|= then
f (3)= | 2(3) +6 |= | 12|= 12
f (-4)= | 2 (-4) + 6|= | -2|= 2
Inverse Function:
• Just as -a is the number for which
a + -a =0= -a + a
• for a≠ 0, a-1
is the number for which
a a-1
= 1 = a-1
a
• In mathematical, g, is uniquely determined by f
and is therefore given name, g=f-1
is real as f
inverse and called the inverse function of f.
• To get the inverse of function by doing the
following
1- replace f (x) = y
2- Interchange x & y.
3- Solve for y
4- Replace y with f-1
Examples:
• F (X)= (X-1)2 Find the F-1
• Solution: let y= (X-1)2
√
√
√
Example:
• F(x)= x+4 Find F-1
Example:
f(x)= x3+3 , Find F-1
Example:
f (x)= 3x+2 Find F-1
Exercise(4)
Find The inverse of
F(x)= 3x+7
G (x) = 5x-3
F (x) = (4x-5)2
Finding domains
Find the domain of the each function.
Solution : we cannot divide by zero , so we must find
any values of x that make the denominator = 0 . these
cannot be inputs. Thus , we set the denominator equal
to 0 and solve for x :
after factoring:
(
so:
then the domain is all real numbers except -1 and 2 .
Find the domain of the function:
√
solution :
the domain is a closed interval starting from 0.5 to infinite
().
Example:
The domain s all the real number except -2, -3
Find the domain of the function :
√
the domain the real numbers, { ∞, -3} U { -2, ∞ }
Exercise(5)
√
Equations
• Linear Equations: Definition:
• A linear Equation in the variable (x) is an equation
that is equivelant to one that can be written in the
form:
• Where:
• , a and b are constants , and a 0
• A linear equation is also called a firt-degree equation
or an equation of the degree one since the highest
power of the variables is (1)
• Solving a linear equation:
• ,eg :
• solve the following;
• Solve :
• Solving a linear Equation:
• Multiply both sides by (4):
(
)
Quadratic Equation:
• The Quadratic Equation in the variable X is an
equation that can be written in the form
• where a, b &c are constant, a ≠ 0
• Solution by factoring:
the solution set is (3, -4).
Solve:
W = 0, or 6w = 5 w= 5/6
the solution set is (0, 5/6).
• (3x-4)(x+1)= -2
• We first multiply the factors on the left side
• 4x- 4x3 = 0
X2+2x-8=0
X2 +2x = 8
by adding 1 to the both side:
√
Exercise(6)
• Solve the following by factoring:
1-
2-
3-
4-
Quadratic formula
√
Solve
• By using quadratic formula
• Solution:
•
√
√
√
Exercise(7)
Solve the following by using Quadratic formula
1-
2-
3-
Systems of Linear Equations
Two – variable Systems :
• Any set of two linear equations is called a set of
equations in the variable x and y or any other
variables .
• The main concern in this section is algebraic
methods of solving a system of linear equations . We
will successively replace the system by other
systems that have the same solutions . We say that
equivalent systems of equations .
• The replacements systems have progressively more
desirable form of determining the solution . Our
passage from a system to an equivalent system will
always be accomplished by one of the following
procedures :
1- Interchanging two equations .
2- Multiplying one equation by a nonzero constant .
3- Replacing an equation by itself plus a multiple of
another equation .
Example:
• consider two equations :
• Solution :
• Multiply equation (1) by 9 and equation (2) by -4 , the
resulted equations are :
• Then by summing the two equations :
• So y = 35 ,
• then by putting y =35 in equation (1) then we find
that x = 40 .
• We can check our answers by substituting x= 40 and
y =35 into both of the ongoing equations . if the
answers are the same , then our solution is true .
this method is called the addition method
Example:
Choose one of the equations , for example equation (1) ,
and solve it for one variable in terms of the other , say x
in terms of y ,then substitute it in the other equation :
From (1) :
In (2) : substitute x, then it will be in the form :
Then y = -1
So x = 5 ,
then investigate whether the solution is correct or not .
In (1) : 5 -3 = 2 (correct)
In (2) : 10+ 4 = 14 (correct)
This method is called the elimination by substitution
Exercise(8)
Solve the systems algebraically :
1-
2-
3-