1. A function relates each element in its domain to a unique element in its range through a rule called the function rule. 2. The domain is the set of possible inputs, while the range is the set of possible outputs. 3. Functions can be represented as ordered pairs in a graph on the x-y plane or through an equation where each input is mapped to a single output.

1. 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.
DOMAIN
-Is the set of all possible value of x in which the corresponding value of y is known
Example
Given y=
RANGE
Is the set of all possible value of y in which the corresponding value of x is known
Example

2. Y=2x
ii. Relation as a mapping
In x- y plane (ordered pair)
FUNCTION:

3. Is the mapping a single element from domain into range?
Not function
TYPES OF FUNCTION
The following are some types of function
1. CONSTANT FUNCTION
f(X)=c
2. LINEAR FUNCTION
f(x)=ax+b
3. QUADRATIC FUNCTION.
f(x) = ax2 +bx+c
4. ABSOLUTE VALUE FUNCTION
f(x)=1x1
5. RATION FUNCTION
1: CONSTANT FUNCTION
SKETCHING THE FUNCTION:

4. Suppose. Given the function
If f(x)=y
Given the function
Suppose

5. y=x for which x for which x>0

6. Solution:
Suppose that
f(x)=y
y=x2-1 x>0

7. 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

8. 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.

9. 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

10. (i) f(x) = x2-1
Its graph
Its graph

11. 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

12. 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.

13. 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

14. 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

15. a = 1, b = -2, c = -8
ASSYMPTOTES
There are lines in which the curve does not touch there are two types for g Assymptotes.
1. Vertical assymptotes.
2. Horizontal assymptotes

16. VERTICAL ASSYMPTOTES(V.A)
Is the one which
HORIZONTAL ASSYMPTOTES
Is the one which
RATIONAL FUNCTION SKETCH

17. Horizontal assymptote (H.A)
Sketch the function

18. Horizontal assymptotes.
Intercepts

19. Sketch.
Intercepts

27. (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

28. y has no restriction: It can be any value
For the Historical A

29. Intercept
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

30. 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)
f(2x) = (2x)2 +1
=4x2+1
2. gof(x) = g f(x) =
=g(x2+1)=
=2(x2+1)
CONCLUSION

31. 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

32. 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

33. -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

34. 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

35. gx=x-1