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-