Mapping & Functionsinstructor: Clement Fisayo (Mr.)

MAPPING & FUNCTIONSINSTRUCTOR: CLEMENT FISAYO (MR.)

ASSISTANT INSTRUCTOR: DOMINIC ASIEDU(MR.)

QUESTION 1:

A function is define by f : x → 4x – 1 on the domain {-3, -2, -1, 0, 1, 2}

x – 2

a)Find the images of the following elements of the domain

(i)0(ii)1(iii)2

b)What elements of the domain correspond to the following images?

(i)4/3(ii)1

c)What value of x makes the function undefined?

d)Determine the range of the function.

QUESTION 2:

A mapping is defined by the rules f : x → 3x + 1.

a)If the domain is {0, 1, 2, 3}, find the range.

b)Use arrow diagram to show whether or not f is one–to-one.

c)If the function of f is to be reshape to x2 + 1 on the domain {-1, 0, 1, 2}, show whether or not f will be one –to-one.

QUESTION 3

Given that f : x is defined as 4x – 1 and g : x = 2x2 + 3

a)Evaluate

g(-3) + f(-2)
g(x) + f(x)
g(x) = f(x)
g(4)/ g(1)

b)If g : x is to be used for the function, state the range for the domain{2, 3, 4, 5}, hence, show whether or not “g” will be one-to-one.

QUESTION 4:

Application of inverse function to resolve any function. i.e. g-1(x), f-1(x), h-1(x)

a)If the function of h of the real variable x is defined as h : x → 2 while f : x = x + 7

x + 4 x – 5

Find the inverse function of h-1 of h
Find the inverse function off-1 of f

b)Hence, evaluate

h-1 (x)
f--1 (-1)

QUESTION 5:

Given that f : x = 2x – 8, find

i)f--1 (x)

ii)f--1 (4)

iii)f--1 (6)

iv)f--1 (-2)

b)2 f--1 (-2)

QUESTION 6:

Composition function can be described as the product of a function substituted one after the other. i.e. the function of

f g(x), gf(x), hg(x), gh(x), fh(x), or hf(x)

If f : x → 2x + 1, g: x → x – 2 while : h : x = 4x – 2, find

a)i. gf(3)ii.fg(3)iii.gh(4)

iv.hf(2)v.fg(3) + f(6)vi.f(6) – fg(3)

b)gf(3)/hf(2)-1

QUESTION 7:

Determine the domain of the function g(x) → x2 + 5

2x2– 3

Find the range of the function
If f : x → 2x + 1, find gof (x) and fog(x)
Find the inverse of g(x) i.e. g-1(x)

If h : x →2x2 + 1 . Find the inverse and range of values of the function.
If f : x →x2 + 2 prove that the function is one-to-one function/

X2– 1

QUESTION 8:

Given f : x → 8x2 – x – 17

(x – 1)(x2 – 3)

Where x is a real number, state the largest possible domain of the function.
Find the inverse of the function if x2 is very small and state range of values of x.
If g(x) → 3x + 2, find gof(2)

QUESTION 9:

f : x → Log2 (2x+ 1) and if f(x)-1= 7/2 find the value of x.
Given f : x → 22x + 1and g: x →x + 1

x - 1

Find gof(x) and prove the inverse of fog(x) = 16

Evaluate f-1og-1(2).

MAPPING & FUNCTIONSSUPPLEMENTARY QUESTIONS

IGBEN MESHACK (MR.)

The function f and g are defined as f : x 4x+3, g: x x/2 (x ≠ 0)

Evaluate:

(i ). f (-1) (ii). g(1/2) (iii). The function (h) is defined on the domain set (1, 2, 3, 4,) by h: x –p 3x-1find the range.

(iv). Given that h: x – p x2 +1 and g: x - p 10x + 1 find (a). h(2) h(-3) h(0), (b). g (2) g(10) g (-3),

( c) . A function is defined by F :x – p 2x – 3 on the domain ( -2 , -1 , 0 , 1, 2 ), find the range of the function, (d). The image of the function defined by F: x – p 2x + 5 is 15 find x

A function is defined by f(x) = 3x – 1 on the domain (-2 , - 1, 0 , 1, 2 )

X -3

i.ind the images of the following elements of the domain (a) 0 (b) 1 (c) 2

ii.What elements of the domain correspond to the following images (a) 7/5 (b) 1 (iii) what value of x makes the function undefined.

iii.Determine the range of the function.

(a). If F(x) = 2x2 – 3x + 4 . simplify f(x) + 3x

(b). if g (x) = (x + 3) (x – 4 )

Find the value of x for which g(x) = 0
Evaluate g(-1)
Find the value of x which g(x) = -3

(c ). Find the inverse function of where

F : x – p 5x – 2 Hence evaluate

F-1(1) ii. F -1(2) (3

4a. If F: x – p 2x g: x – p x – 3

Find (1 ) gf(3) and fg (3)

g -1 f -1(3) and f – 1 g – 1 (3)
g -1 f – 1(0) and F-1 g-1(0)

(B). Given that F: x – 0 3x

g: x – p x – 5

h: x – p 2x + 1

find (I ). F – 1, (ii). g -1 , (iii). h – 1 ,( iv). g -1 h -1 (v). F -1 h -1

5 . The function f and g of the real variable x are defined

F: x – p 1 – 2x and g: x – p 1/x + 3

State the domain of each function.
Show that g (a) = g(b) = p a = b. what conclusion can you draw from this result.
Find the inverse function g -1 of g and hence the value of x for which f g (x) = g -1 (x) leaving your answer in sure form.

(b) . Under the mapping h(x) = px2 - 9x + 2, the image of 3 is 14 and the image of -2 is 24. Find

The value of p and q
The elements whose image is 4.

ELEMENTARY STATISTICSINSTRUCTOR: ABDOULIE B. CONTA (MR.)

ASSISTANT INSTRUCTOR: ABDOU SECKA(MR.)

QUESTION 1:

The data below gives the marks obtained by 30 students in a science class test at Bason Upper Basic School.

2, 2, 4, 3, 2, 4, 0, 4, 5, 1

3, 3, 4, 0, 3, 4, 5, 5, 4, 6

1,5 ,3 ,2 ,6 ,2 ,5 ,4 ,3, 4

Prepare a frequency distribution table for the data.
Find the mean mark
What is the model mark
What is the probability that a student chosen from the class obtained a mark less than 5?

QUESTION 2:

In an objective test marked out of 40, the marks scored by 35 students are given in the table below:

Marks Scored / 3 / 8 / 13 / 18 / 23 / 28 / 33 / 38
No Of Students / 2 / 5 / 9 / 6 / 3 / 4 / 5 / 1

Find the probability that a students selected at random from the class scored

Marks greater than 28
Marks less than 28
33 marks
Between 3 and 33 marks.

QUESTION 3:

The following data gives the monthly budget of a man.

Items / Amount used (D)
Food / 100
Rents / 50
Savings / 50

Draw a pie chart to illustrate the information.
What fraction of the monthly budget is spent on rent?

QUESTION 4:

Two groups take the same math’s test. The test is marked out of 50.

Ebrima writes down the marks for his group.

25, 47, 49, 31, 38, 24, 19, 22, 38, 25.

Calculate the mean mark for his group.
What is the range of marks for his group? Binta writes down the mark for her group.

The lowest mark is 12.

The range is 30.

What is the highest mark for her group?

The mean for her group is 25.

Compare the results of the two groups.

QUESTION 5:

A survey was carried out on the shoe sizes of 25 men. The results of the survey were:

106989

86799

779109

89858

891087

a. (i)What is the mode of the shoe sizes?

(ii)What is the median of the shoe sizes?

b. Draw a histogram to shows the results of the survey.

QUESTION 6:

In a factory, there are 79 male and 74 female managers. Managers can be either junior or senior. There are 28 male senior managers.

(a)Construct a bar chart to show the number of male and female managers in the junior and senior management.

(b)Show the information in 3(a) in a pie chart.

(c)Comment on the proportion of women in junior and senior management.

QUESTION 7:

Eight people work in the accounts department of a company. The manager of the department is paid €260 a week. The assistant manager gets €220 a weeks, and the assistant manager’s secretary gets €100. There are four clerical staff, each of whom gets €90 a week.

One of the clerical staff complains to her track union that people in the accounts department are poorly paid. The manager tells the union that the average (mean) weekly pay in the department is €132.50.

a)Show that the manager is right in what he says, by doing the calculation yourself.

b)Why is the mean weekly pay misleading in this case?

PROBABILITYASSISTANT INSTRUCTOR: DAUDA ANDREW CONTEH

QUESTION 1:

The probability that a Malaria Patient (M) survives when administered with a newly discovered drug is 0.27 and the probability that a typhoid patient (T) survives when injected with another newly discovered drug is 0.85. What is the probability that:

Either of the two patients survive?
Neither of the two patients survive?
At least one of the two patients survive?

( Give answers correct to 2 significant figures)

QUESTION:

2a. X and Y are two independent events and that P(X) = 0.5 and P(Y) = a and P(XUY) = 0.8, Find the

Value of a.

2b. A bag contains 12white balls and 8 black balls.

Another bag contains 10 white balls and 15 black balls .

If two balls are drawn without replacement from each bag, find the probability that:

All four balls are black
Exactly one of the four balls is black.

QUESTION:

3a. A coin is tossed twice and die is rolled once.

Find the probability of the following events.

Two heads and a 5
Exactly one head and a 1 or 2
At least one head and not a 2

3b. A and B are two independent events such that

P(A)=3/7 and P(AnB)=2/5, find:

P(B)
P(AuB)

QUESTION:

4a. For every ten person in a city, one is left-handed. If Six persons are selected at random

From the city, find the probability that:

Exactly 3 are left-handed.
More that half are left-handed.
At least 5 are left-handed.

VARIATIONSINSTRUCTOR: BRIMA B. CONTEH (MR.)

U 3 x , U = 9 when x = 27

y and y = 2

Find the relation between u , x and y
Find u when x = 64 and y =12

A B when B= 36 and C = 16, A = 27

Find the formula which connects A, B and C
Find A when B = 33 and C = 18

X is partly constant and partly varies as y .

When y = 4 , x = 60 and when y = 12, x = 100

Find the relationship between x and y
Find x when y = 6

The cost of a car service is partly constant and partly varies with the time it takes to do the work. It costsGMD35.00 for 5 ½ hour service and GMD29.00 for a 4 hour service.

Find the formula connecting cost ( C ) with time T hours
Hence find the cost of a 71/2 hour service.

VARIATIONSINSTRUCTOR: BRIMA B. CONTEH (MR.)

QUESTION 1:

A car uses 25 liters of petrol for a journey of 175km. How many liters will it use for a journey of 275km?

QUESTION 2:

Ten men weed a field in 16 days. Howlong would it have taken 8 men to weed the field?

QUESTION 3:

P Q and P = 4.5 when Q = 12.

Find (a) the relationship between P and Q.

(b) P when Q = 16 (c) Q when P = 2.4

QUESTION 4:

If a car travels for a certain time, the distance travelled variesdirectlywith the average speed, when the average speed is 35km/hr, the distance travelled is 115 km. What distance is travelled when the average speed is 64km/hr?

QUESTION 5:

A 1 and b = 8 when h= 57

(a) the relationship between b and h

(b)Find (i) b when h = 4 (ii) h when b = 1

QUESTION 6:

If a post which is 5m high cast a shadow which is 12m long, how long will the shadow of a 9m post be?

QUESTION 7:

A man travels a distance of 80km (a) Copy and completes the table below

Time / 6 / 4 / 2
Speed (km/hr)
1/ Speed

(b) Draw a graph connecting time with the reciprocal of speed

(c.) Use your graph to find (i) speed if the time taken was 5 hours

(ii) the time at a speed of 25km/hr

CONSTRUCTIONS INSTRUCTOR: ISSA DUMBUYA (MR.)

Construction of basic angles, line segment using pair of compasses, ruler, protractor etc

Line segment
Bi-section of a given line divided it into two, four equal parts etc
Measuring two points from a given line segment in mm, cm , etc
To construct 135, 105, 90, 60, 75, 45, 50, 22 ½ , 30, etc

Construction of

Triangles
Quadrilateral
Bi-section of angles
Measuring angles using protractor.

PRACTICALS

Using a ruler, pencil and a pair of compasses only construct a:

(a)Triangle XYZ in which XY = 55mm, XY = 82mm, XYZ = 750and XYZ = 450

(b) Bisect angle XYZ and XZY

(d) Measure (i) WZ

Quadrilateral PQRS in which PQ = 5cm, QR = 7cm, SPQ = 900 , PQR = 1200,QRS = 450

(a)Triangle DEF in which DE = EF = 65mm, DF = 56mm

(b)What figure have you constructed?

(c)Bisectall the angles of the triangle DEF and let the bisectors meet at G

(d)Measure angle EGD

Quadrilateral BCDE in whichBC = 7cm, CD = 5cm, and EBC = 900, BCD and CDE = 600

Measure (i) BED (ii) DE

Quadrilateral KLMN in which NKL = 900 , KLM = 450, KL = 5.6cm. Measure KN.

Which quadrilateral have you constructed?

MANAGEMENT OF MATHEMATICS DEPARTMENT

INSTRUCTOR: DAUDA ANDREW CONTEH

How would you manage the department as a head if:

a)There are no materials such as chalk, board protractor, compasses or Set Square for

b)Technical Drawing or Mathematics lessons?

c)Textbooks for teaches in the department are insufficient?

d)There are insufficient desks / benches for students?

e)The class size is high (say more than 40 students as it is in some schools)?

How would you deal with any teacher who sees you not suitable as a head of department or sees you as too harsh?
What would be your stand if there is a misunderstanding between teachers in then department?
How would you deal with a difficult Principal or Proprietor / Proprietress of you school?
How would you deal with lateness of teachers to classes?

Youth Care Foundation, The Gambia / Centre for Education In Mathematics and Computing, Univ. of Waterloo, Canada

MANAGEMENT OF MATHEMATICS DEPARTMENT

INSTRUCTOR: DAUDA ANDREW CONTEH

How would you manage the department as a head if:

a)There are no materials such as chalk, board protractor, compasses or Set Square for Technical Drawing or Mathematics lessons?

b)Textbooks for teaches in the department are insufficient?

c)There are insufficient desks / benches for students?

d)The class size is high (say more than 40 students as it is in some schools)?

How would you deal with any teacher who sees you not suitable as a head of department or sees you as too harsh?
What would be your stand if there is a misunderstanding between teachers in then department?
How would you deal with a difficult Principal or Proprietor / Proprietress of you school?
How would you deal with lateness of teachers to classes?

Youth Care Foundation, The Gambia / Centre for Education In Mathematics and Computing, Univ. of Waterloo, Canada

GEOMETRYINSTRUCTOR: KOLAPO ABDUL (MR.)

1. The figure below shows a machine part made from

two cones of base radius 3cm. The height of the cones are 10cm and 4cm,

a. Calculate the volume of the machine part in cm3.

b. If the machine part is made of brass of density 8g/cm3,calculate its mass in kg.

2. A cuboid is divided into two equal right-angle triangleprism.

Calculate the volume of the show prism.

3. If the volume of a rectangular-based pyramid is 70cmand the base area is 28cm2, calculate the height of the pyramid.

4. Calculate the shaded area of the following.

TECHNICAL COMMUNICATION

COURSE OUTLINE:

‐ The course will highlight appropriate communication skills in mathematics classroom.

‐ Teacher will be taken through technical terms: simply, Evaluate, find the value of…., calculate,

Express, solve, locate, Identify, find the least, at most, estimate etc.

‐ Teachers and instructor will look into some technical questions and solve them accurately.

Some Technical Problems

1) From two points on opposite side of a pole 33m high, the angles of elevation of the top of the

pole are 530 and 670. If the two points and the base of the pole are on the same horizontal

level, calculate correct to 3 significant figures the distance between the two points.

2) (a) A man saved Le 3000 in bank P, whose interest rate was X% per annum and Le 2000 in

another bank Q whose interest rate was Y% per annum. His total interest in one year was le 640.

If he had saved Le 2000 in bank p and Lle3000 in bank Q for the same period, he would have

gained le 20 as additional interest. Find the values of x and y

(b) A man invested D20, 000 in bank A and D25, 000 in bank Bat the beginning of a year. Bank A

pays simple interest at a rate of Y% per annum and B pays 1.5y% per annum, if his total interest

at the end of the year form the two banks was D6400, find y

3) (a) A motorcycle travels a distance of 240km at a uniform speed. If it had been 4km/hr slower, it

would have taken 2 hour more to cover the distance. Find its speed.

3) (b) In Canada, they measure distance in kilometers. One kilometer is about 60% of one mile.

Estimate this same speed in:

(i) Meters per hour

(ii) Meters per second.

4) If the equation px2 + (p+1) x +p =0 has equal roots, identify the value of p for which p>0 (p is

greater than zero)

5) A hemispherical tank of diameter which is 10m is filled by water issuing from a pipe of radios

20cm at 2m per second. Calculate, correct to 3 significant figures, the time, in minutes, it takes

to fill the take.

6) (a) Two fair dice are tossed together once. Find the probability of getting a total:

(i) Of at least 9,

(ii) Less than 4

(iii) At most 4

MODULAR ARITHMETIC INSTRUCTOR: K.A GYAN AKWANDA (MR.)

Introduction:

(i) Clock arithmetic is a typical example of modular arithmetic. In clock arithmetic every whole number is equivalent to a whole number less than 12.

In modulus 12 , every whole number is equivalent to one of 0,1,2, ….. , 11. Any multiple of 12 is equivalent to 0. Thus, 24 = 0 (modulus 12)

In clock arithmetic 6 + 7 =m1 (mod 12) i.e.we take the reminder after division by 12. In modular arithmetic, we use and for addition and multiplication respectively. Thus, 6 7 = 1 (mod 12) and 4 3 = 0 (mod 12)

Generally, to do arithmetic, in modulus n we do the ordinary arithmetic and if the result is more than n, find the remainder on division by n. The result would be in the set 0,1,2, ….., n-1

(ii) Week arithmetic ( using the days of the week) is arithmetic modulus 7. Here any whole number is equivalent to one of 0, 1, 2, ….., 5. Any multiple of 7 is equivalentto 0.

Modular Table

Modular arithmetic can beshown in combination tables under addition and multiplication .

For example, for addition modulus 5, the results are from set 0,1,2,3,.4.

/ 0 / 1 / 2 / 3 / 4
0 / 0 / 0 / 2 / 3 / 4
1 / 1 / 2 / 3 / 4 / 0
2 / 2 / 3 / 4 / 0 / 1
3 / 3 / 4 / 0 / 1 / 2
4 / 4 / 0 / 1 / 2 / 3

QUESTION 1:

Construct and tables for modulus 6. Use the tables to solve the following equations

in arithmeticmodulus 6.

i) x x = (mod 6)

ii) x 4 = ( mod 6)

iii) 2 x = (mod 6)

iv) x x = (mod 6)

i) 3 x = (mod 6)

ii) x x = (mod 6)

QUESTION 2:

Construct the combination tables for the set 0,1, 2, 3,4, under addition (mod 5) and the set

1, 2, 3, 4 under addition (mod 5).

Use the tables to solve the following equations in arithmetic modulus 5.

i)x x = 1 (mod 5)

ii)x 3 = 2 ( mod 5)

iii)2 x = 1 (mod 5)

iv)x x = 4 (mod 5)

v)x x = 2 (mod 5)

vi)2 = x 4

QUESTION 3:

Construct and tables for modulus 7 on the set 0,1, 2, 3,4, .

(a)Use the tables to evaluate the following in the arithmetic modulo 7.

i)(2 2) (3 1)

ii)(2 4) (3 3)

(b)Use the tables to solve the following equations in arithmetic modulus 7.

i)(x x) = 2

ii)x ( x 4) = 5

(c)x (x 4) = 0

Youth Care Foundation, The Gambia/ Centre for Education in Mathematics and Computing, Univ. of Waterloo, Canada.