**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 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

Y=2x

ii. Relation as a mapping

In x- y plane (ordered pair)

FUNCTION:

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) = ax^{2} +bx+c

4. ABSOLUTE VALUE FUNCTION

f(x)=1×1

5. RATION FUNCTION

1**: CONSTANT FUNCTION**

SKETCHING THE FUNCTION:

Suppose. 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

y=x^{2}-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 **ax ^{2} + bx + c**

Where a, b and c are real numbers and a ≠ 0

**Example**

Draw the graph of the function

(i) f(x) = x^{2}-1

(ii) f(x) = -x^{2}-1

Solution

Table Value

(i) f(x) = x^{2}-1

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) = ax ^{3 }+ bx^{2} + cx + d**

Where a, b, c and d are real numbers and a ≠ 0

**Example**

Draw the graph of the following function

f(x) = x^{3} – 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) = **ax ^{2} + 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)** = **ax ^{2} + bx + c**

Factorizing out the constant a

**Example**

Sketch the graph of f(x) = x^{2} + 2x+ 8, determine the turning point and the intercepts

**Solution**

x^{2} + 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 x^{2} + 2x+ 8= 0 to ax^{2} + 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

Horizontal assymptote (H.A)

Sketch the function

Horizontal assymptotes.

Intercepts

Sketch.

Intercepts

(y-1) x^{2}-2(y-1)x-3(y-1)=-4x+8

(y-1)x^{2}-2(y-1)x+4x-3(y-1)-8=0

(y-1)x^{2}-2yx+2x+4x-3y+3-8=0

(y-1)x^{2} +(-2y+6)x-(3y+5)=0

For real value of x

b^{2}-4ac ≥ 0

(-2y+6)^{2} +4(y-1) (3y+5)≥0

(4y^{2}-24y+36)+ (12y^{2}+8y-20)

16y^{2} – 16y +16 ≥0

y^{2}-y+1>0

y has no restriction: It can be any value

For the Historical A

Intercept

2xy -3y= 4x^{2} + 8x-5

4x^{2} +8x-2xy-5+3y

4x^{2} (8-2y)x +(3y-5)=0

For the real value of x

b^{2}-4ac ≥ 0

(8-2y)^{2}-4.4(3y-5)≥0

64-32y+4y^{2}-48y+80≥0

4y^{2}-80y+144≥0

y^{2}-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) = x^{2}+1 and g(x)

=2x.

Find (i) fog(x)

(ii).gof(x)

Approach f(2x) =2(x^{2}+1)

1.fog(x) = f (g(x)

f(2x) = (2x)^{2} +1

=4x^{2}+1

2. gof(x) = g f(x) =

=g(x^{2}+1)=

=2(x^{2}+1)

CONCLUSION

fog gof, hence the compacite function is not commutative

ASSOCIATIVE PROPERTY

Given

F(x)=x^{2}-1, g(x)=3x and h(x) =2/x

(i)(fog) oh

(ii)fo (goh)

fog=f (gx)=f(3x)=(3x)^{2}-1

9x^{2}-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

Mathematics (from Ancient Greek μάθημα; máthēma: ‘knowledge, study, learning’) is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

Most mathematical activity involves the use of pure reason to discover or prove the properties of abstract objects, which consist of either abstractions from nature or—in modern mathematics—entities that are stipulated with certain properties, called axioms. A mathematical proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration.

Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein’s general relativity, which replaced Newton’s law of gravitation as a better mathematical model.

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