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FUNCTION AND RELATION
FUNCTIONS:
Is the corresponding between two objects. E.g older than. Smaller than ect.
Relation can be thought as:
(i) Rule
(ii) A mapping
Example
(iii) A graph of x-y plane.
(iii) A graph of x-y plane.
DOMAIN
-Is the set of all possible value of x in which the corresponding value of y is known
Example
Given y=
Given y=
RANGE
Is the set of all possible value of y in which the corresponding value of x is known
Example
Y=2x
ii. Relation as a mapping
ii. Relation as a mapping
In x- y plane (ordered pair)
FUNCTION:
Is the mapping a single element from domain into range?
Not function
Not function
TYPES OF FUNCTION
The following are some types of function
1. CONSTANT FUNCTION
f(X)=c
2. LINEAR FUNCTION
1. CONSTANT FUNCTION
f(X)=c
2. LINEAR FUNCTION
f(x)=ax+b
3. QUADRATIC FUNCTION.
3. QUADRATIC FUNCTION.
f(x) = ax2 +bx+c
4. ABSOLUTE VALUE FUNCTION
4. ABSOLUTE VALUE FUNCTION
f(x)=1×1
5. RATION FUNCTION
5. RATION FUNCTION
1: CONSTANT FUNCTION
S
KETCHING THE FUNCTION:
KETCHING THE FUNCTION:
Suppose. Given the function
If f(x)=y
Given the function
If f(x)=y
Given the function
Suppose
y=x for which x for which x>0
Solution:
Suppose that
f(x)=y
f(x)=y
y=x2-1 x>0
Step function
Sometimes referred as compound function, are linear function whose variables have a special relationship under certain conditions that make their graphs break in intervals(Look like steps).To understand the concept, let us look at the following example.
The cost of shaving the hair of different age-groups in a central salon are as follows
a) Shaving the group against ten to twenty years costs Tsh 2000/=
b) The group aging between twenty and thirty exclusive costs Tsh 4000/=
c) The group aging thirty and above costs Tsh 6000/=
From the above information provide domain and range
Solution
If we let x the ages and f(x) be the costs, then we can interpret this problem as a step function defined by
The domain of this function is a set of real numbers such that x ≥ 10.
The range of this function is {200,400,600}.
Graph of Quadratic function.
A quadratic function is a polynomial of the second degree.
It is a function of the general form ax2 + bx + c
Where a, b and c are real numbers and a ≠ 0
Example
Draw the graph of the function
(i) f(x) = x2-1
(ii) f(x) = -x2-1
Solution
Table Value
(i) f(x) = x2-1
Its graph
Its graph
Its graph
Drawing graph of cubic function
-When the polynomial function is reduced to the third degree a cubic function is obtained.
The cubic function is take a general form f(x) = ax3 + bx2 + cx + d
Where a, b, c and d are real numbers and a ≠ 0
Example
Draw the graph of the following function
f(x) = x3 – 9x
-The intercept are points (-3,0),(0,0),(3,0)
-There are two turning points; the maximum i.e (-2,10) and the minimum i.e (2,-10)
-The domain is the set of all numbers
-The range is the set of all real number’s y.
For the turning point let us consider the function f(x) = ax2 + bx + c .b. The function f may be expressed in the form of g a[g(x)] + k
Where g(x) is another function in x and k is a constant as follows.
f(x) = ax2 + bx + c
Factorizing out the constant a
Example
Sketch the graph of f(x) = x2 + 2x+ 8, determine the turning point and the intercepts
Solution
x2 + 2x+ 8= 0
Solving we get
(x + 2)(x – 4) = 0
x + 2 = 0 x – 4 = 0
x = 2 , x = -4 which are intercepts
-The y-intercept C is -8
-To obtain the turning points, equate x2 + 2x+ 8= 0 to ax2 + bx + c = 0, so that the comparison we get
a = 1, b = -2, c = -8
ASSYMPTOTES
There are lines in which the curve does not touch there are two types for g Assymptotes.
- Vertical assymptotes.
- Horizontal assymptotes
VERTICAL ASSYMPTOTES(V.A)
Is the one which
HORIZONTAL ASSYMPTOTES
Is the one which
RATIONAL FUNCTION SKETCH
Is the one which
RATIONAL FUNCTION SKETCH
Horizontal assymptote (H.A)
Sketch the function
Horizontal assymptotes.
Intercepts
Sketch.
Intercepts
(y-1) x2-2(y-1)x-3(y-1)=-4x+8
(y-1)x2-2(y-1)x+4x-3(y-1)-8=0
(y-1)x2-2yx+2x+4x-3y+3-8=0
(y-1)x2 +(-2y+6)x-(3y+5)=0
For real value of x
b2-4ac ≥ 0
(-2y+6)2 +4(y-1) (3y+5)≥0
(4y2-24y+36)+ (12y2+8y-20)
16y2 – 16y +16 ≥0
y2-y+1>0
y has no restriction: It can be any value
For the Historical A
For the Historical A
Intercept
2xy -3y= 4x2 + 8x-5
2xy -3y= 4x2 + 8x-5
4x2 +8x-2xy-5+3y
4x2 (8-2y)x +(3y-5)=0
For the real value of x
b2-4ac ≥ 0
(8-2y)2-4.4(3y-5)≥0
64-32y+4y2-48y+80≥0
4y2-80y+144≥0
y2-20y +36≥0
(y-2) (y-18)≥0
Condition
(y-2)≥0 y-18≥0
(y-2)≤2, y-18≤0
y ≥ 2, y ≥18
y ≤ 2, y ≤18
Function can not lie between 2 and 18
COMPOSITE FUNCTION.
Two functions f and g are said to be composite function of fog= f(g) (x)
NOTE: COMMUTATIVE PROPERTY
Given f(x) = x2+1 and g(x)
=2x.
Find (i) fog(x)
(ii).gof(x)
Approach f(2x) =2(x2+1)
1.fog(x) = f (g(x)
1.fog(x) = f (g(x)
f(2x) = (2x)2 +1
=4x2+1
2. gof(x) = g f(x) =
2. gof(x) = g f(x) =
=g(x2+1)=
=2(x2+1)
CONCLUSION
fog gof, hence the compacite function is not commutative
ASSOCIATIVE PROPERTY
Given
F(x)=x2-1, g(x)=3x and h(x) =2/x
(i)(fog) oh
(ii)fo (goh)
fog=f (gx)=f(3x)=(3x)2-1
9x2-1
Since fo(goh)=fo(goh) hence the compacite function is associative property
FUNCTION
A f unction is a function when the line parallel to the y-axis cuts only once on the curve.
The line parallel to the x-axis cuts the curve only
-An inverse function is the one which each elements from Domain matches exactly in range conversely each element from range matches exactly with Domain
Given f(x)=2x-1
Find f-1(x)
Approach
Sketch
(i) f(x) – state its Domain
(ii)f-1 (x)
soln
f(x)=x+1
suppose f-1(x) = g(x)
fog=f(gx)=x
gx+1=x
gx=x-1
