MATHEMATICS NOTES FORM FIVE:FUNCTION AND RELATION

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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
E:\..\..\..\thlb\cr\tz\__i__images__i__\011.png

(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=  E:\..\..\..\thlb\cr\tz\__i__images__i__\uuuuuu1.jpg

E:\..\..\..\thlb\cr\tz\__i__images__i__\hioo.PNG

RANGE

Is the set of all possible value of y in which the corresponding value of x is known

 Example

Y=2x
E:\..\..\..\thlb\cr\tz\__i__images__i__\ma1.png

ii.     Relation as a mapping
E:\..\..\..\thlb\cr\tz\__i__images__i__\hgo.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\iu.png

In x- y plane (ordered pair)
E:\..\..\..\thlb\cr\tz\__i__images__i__\qqqqqqqqqqqqq.png

FUNCTION:

Is the mapping a single element from domain into range?
E:\..\..\..\thlb\cr\tz\__i__images__i__\hds.png

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)=1×1

5. RATION FUNCTION

  E:\..\..\..\thlb\cr\tz\__i__images__i__\TTTTTTTTTTTTTTTT.png

1: CONSTANT FUNCTION

SKETCHING THE FUNCTION:

Suppose. Given the function
E:\..\..\..\thlb\cr\tz\__i__images__i__\ma2.png
If f(x)=y
E:\..\..\..\thlb\cr\tz\__i__images__i__\phoneee1.PNG

E:\..\..\..\thlb\cr\tz\__i__images__i__\ninnnnn.PNG

Given the function
E:\..\..\..\thlb\cr\tz\__i__images__i__\pppppppppppppppppppp.png

  Suppose

   y=x for which x for which x>0

    E:\..\..\..\thlb\cr\tz\__i__images__i__\ma41.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\leooo1.PNG

  E:\..\..\..\thlb\cr\tz\__i__images__i__\sawaaa.PNG

    Solution:

Suppose that
f(x)=y

y=x2-1  x>0

E:\..\..\..\thlb\cr\tz\__i__images__i__\coppp.PNG

  E:\..\..\..\thlb\cr\tz\__i__images__i__\involve.PNG

  E:\..\..\..\thlb\cr\tz\__i__images__i__\ma9.png

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
E:\..\..\..\thlb\cr\tz\__i__images__i__\onee4.PNG

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

E:\..\..\..\thlb\cr\tz\__i__images__i__\save2.PNG

     Its graph
E:\..\..\..\thlb\cr\tz\__i__images__i__\dogooo.PNG

E:\..\..\..\thlb\cr\tz\__i__images__i__\nomasanaa.PNG
Its graph
E:\..\..\..\thlb\cr\tz\__i__images__i__\graphhhh.PNG

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

E:\..\..\..\thlb\cr\tz\__i__images__i__\masterrr.PNG

-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

  E:\..\..\..\thlb\cr\tz\__i__images__i__\kichwaaaa.PNG

E:\..\..\..\thlb\cr\tz\__i__images__i__\rexall.PNG

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
E:\..\..\..\thlb\cr\tz\__i__images__i__\thus2.PNG

E:\..\..\..\thlb\cr\tz\__i__images__i__\daaaa1.PNG

ASSYMPTOTES

There are lines in which the curve does not touch there are two types for g Assymptotes.

  1. Vertical assymptotes.
  2. Horizontal assymptotes

E:\..\..\..\thlb\cr\tz\__i__images__i__\azx.png

VERTICAL ASSYMPTOTES(V.A)

Is the one which
E:\..\..\..\thlb\cr\tz\__i__images__i__\ma10.png

HORIZONTAL ASSYMPTOTES
Is the one which

E:\..\..\..\thlb\cr\tz\__i__images__i__\ma11.png

RATIONAL FUNCTION SKETCH
E:\..\..\..\thlb\cr\tz\__i__images__i__\EEEEEEEEEEEEEEEEEEEEEEEEEEEWW.png

Horizontal assymptote (H.A)

E:\..\..\..\thlb\cr\tz\__i__images__i__\iiiiiiiiiiiiiiiiiiiiiiiiiiii.png

    E:\..\..\..\thlb\cr\tz\__i__images__i__\ds2.png
E:\..\..\..\thlb\cr\tz\__i__images__i__\lo.png

Sketch the function

E:\..\..\..\thlb\cr\tz\__i__images__i__\HHHHHHHHHHHHHHHHHHHHHHHOO.png
Horizontal assymptotes.
E:\..\..\..\thlb\cr\tz\__i__images__i__\99999999999999999.png

Intercepts

E:\..\..\..\thlb\cr\tz\__i__images__i__\ma12.png

Sketch.
E:\..\..\..\thlb\cr\tz\__i__images__i__\xaaaaaaaa.PNG

  E:\..\..\..\thlb\cr\tz\__i__images__i__\bt.png

Intercepts
E:\..\..\..\thlb\cr\tz\__i__images__i__\yyyyyyyyyyyyyyyyy.png
E:\..\..\..\thlb\cr\tz\__i__images__i__\uuuuuuuuuuuuuuuuuuuuu.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\tayarrrr.PNG

  E:\..\..\..\thlb\cr\tz\__i__images__i__\0000000000000.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\4444444444444444.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\nomaaa.PNG

  E:\..\..\..\thlb\cr\tz\__i__images__i__\ssssssssssssssssss.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\hhhhhhhhhhhhhhhh.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\MMMMMMMMMMMM.png

   E:\..\..\..\thlb\cr\tz\__i__images__i__\ma15.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\ma16.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\jjjjjjjjjjjjjj.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\dddddddddddd.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\ma17.png
E:\..\..\..\thlb\cr\tz\__i__images__i__\duhhh.PNG
E:\..\..\..\thlb\cr\tz\__i__images__i__\ma181.png
(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

  E:\..\..\..\thlb\cr\tz\__i__images__i__\ma19.png

y has no restriction: It can be any value
E:\..\..\..\thlb\cr\tz\__i__images__i__\tttttttttttttttttttt.png
For the Historical A
E:\..\..\..\thlb\cr\tz\__i__images__i__\999999999999999.png

Intercept
E:\..\..\..\thlb\cr\tz\__i__images__i__\cfc.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\vf.png
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

E:\..\..\..\thlb\cr\tz\__i__images__i__\xxxxxkkkkk.png

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

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

E:\..\..\..\thlb\cr\tz\__i__images__i__\chinnn.PNG

E:\..\..\..\thlb\cr\tz\__i__images__i__\ma20.png

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.

E:\..\..\..\thlb\cr\tz\__i__images__i__\desaaa.PNG
The line parallel to the x-axis cuts the curve only
E:\..\..\..\thlb\cr\tz\__i__images__i__\HF.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\BW.png

 -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

E:\..\..\..\thlb\cr\tz\__i__images__i__\ma21.png
E:\..\..\..\thlb\cr\tz\__i__images__i__\8888.png
E:\..\..\..\thlb\cr\tz\__i__images__i__\given.PNG

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
E:\..\..\..\thlb\cr\tz\__i__images__i__\ma231.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\hb.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\ma24.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\yt1.png

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