1969-02-13 — Page 24

育教僑華頁三第張六第 日七廿月二十年申戊夏WAH KIU YAT PO 報日僑華

四期星日三十月二年九六九一履公年八十五國民藏中

健

文中學會考試題預習

1969

日文中學會考試題預習

Vane

現代數學科

生物科 (十五)

百

-Rachis

(十五)

BIULLOY (15)

MODERN MATHEMATICST( 15 )

Different types of functions

"Let f be a function from A into B. f is called one-one function if there are no two different. elements in A have the same image in B. That is, f(a) = f(b) implies a = b. or a / b implies f(a) ff(b).

Examples

(1) Let [:D→→→→ D be defined by f(x)**

Answer to the exercise of last week:

Ans

1. fis

a one-one function, because different x's have different f(x)'a.

(2) Let :DD be defined by f(x) = x2. Then fisi

not a one-one function, because for two values of x such as x = 2, x = -2, f(x) = 4.

Let f be a function from A into 8. £ is called

an onto function if every element in B is an image of at least one element of A

Examples:

(3) Let f:A-96, where A = {1,2,3,4)

B - {0,1,2,3], be defined by f(x) =

Then I is an onto function, because every element)

in 5 is an image of an element in A..

Let fiA—>Ę, where A {-1,1,-2,2, -33 and

B = {1,4,9} be defined by f(x) = x2, f is an

onto function, because every element in B la the image of two elements in A.

A function may be both one-one and onto, such as the Example (3) above.

The

three different functions are best illustrated by diagrams of mappings as follows,

10 0000

onto function

one-ore Junction

fone-one and onto

function.

10. Operations with function and composite function

Iff and g are real functions with dom ins Dr and Dg, then the definitions of additions, subtraction, multiplication, and division of the functions are defined as follows,

(1) By the sum of ƒ and g we mean there is

function h such that A

h(x) = f(x) = g(x) and the domain or,

h = DAT

}[ By the difference of f and a we mean there 19

function h such that

h(x) = f(x) = g(x) and the domain org h = Dp A Dgx

(3)By the product of f and g, we mean the function

h such that h(x) = f(x).g(x), where the domain

(4) By the quotient of f and g, we mean the function

b, such that h(x) = {(2)

s(x) 70 and

domain is DrDg

Example:

Suppose f(x)=3x-2 and g(x)”-

and the domains for each is the set of real

numbers. Then

L(x) +j8(x) = 3x*= 2 + (2x2)-

f(x)2− g(x) = 3x

2

-3x)(3x

If fland g are real functions with domains Dryand Dg, then the composite function gif) means the function h such that h(x) = g[f(x)] and

Dh = {x & Dp / f(x)E U_}. Notice that if the range off and the domain of g have no common element,

then h will be the empty

g(f) and f(g) are not the

Examples

(1) Let f be defined by f(x)=

defined by

Notice also that)

and g be

(a) trive a labelled diagram of a bird, showing

its chief external features. by what ways a bird specially adapted for flight? bbith the help of diagrama, describe briefly,

the structure's of a wing feather of the bird.

Nostril

Beak

Neck

Thorax Abdomen

Quill

Head Eye Ear

Rachis

Back Wing feathers

-Tail

Leg

The External Features of A pigeon

The ways by which a bird is adapted for! flight may be considered under a number of headings:

(1) The body 13 streamlined so that the air flow smoothly past as the bird fliesk ii) The strong breast bone or keel, to which

are attached powerful pectoral muscles which make up much of the bird's weight, (114). The body is covered with light feathers.

instead of scales; the bones are mostly hollow, and these reduce the density of the body.

(iv) The body temperature of the oira 19 high,

and this presumably means a high rate of

with the powerful musclár action.

The fore-limbs are modified into wings with feathers which are light and constructed to facillated flight. The small feathers are very good insulators by virtue of the layer of air which they entrap, and so they help to keep up body temperature. The joints of the skull are entirely lost and the face prolonged to form a beak There is only one peg at the junction or the head and atlas, so that the bird can turn it head almost completely round The transparent third eyelid plays an important part in protection of the eve whilst the bird is in flight.

11 The lungs are well-developed, a

connecceo

with them are air-sacs which extend a long the body-contents and into some of the bones. These will ensure that the well supply of air is maintained.

1x) The intestinal region is relatively snore

compared to the intest ine of mammals This helps to prevent too much waste substances accumulate in the body thus reducing the body weight.

A

A large volume or excretory produets can be removed, chiefly as uric acid which can be eliminated with little loss of water- This also helps to reduce the body weight, The bird has lost the urinary bladder add the right ovary and oviduct, possibly as one means of minimizing its weight.

ving feather has a median solid shaft known as the rachis, at the base of which is the hollow quill. The lower tip of the quilt is a 11. opening, the inferior umbilicus. ravond

De central shaft is the wide part of the feather,

the vane. The vane cons 1819 of numerous fine barbs emerging on each side. barb has smaller barbules branching from t the lower carbules carry books which inter Leak with barbules of the

flat vane of the raih Series. Itus, the

is formed.

Diagrams (a) and (b) qerine a runction, but diagram (b) does not, because in (b) not each element in A has an image.

e(r) f(g)

(2) Given f

ana

find g(f)

(2) let

and f

8(1)

[f(a)

(3) Given:

f(g).

B(f) = (x3

(2)3

11.Worked examplesk

find g(f) and

(1) Indicate which of the following define a

function of A into B if A = {1,2,4,5] and B={1,3,5,7

00 00 00

(D)

defined by

if x>2

((x) -(x2 - \ -23, rima 1(3), I(−2), r(0)

Solutionte

1(3) × 3—2:37=13 r(-2) ~2+1 = -1 (0) 1011:611

(3) Let A [a,b] and B = {1,2,3), now many

different functions are there from A into Br Solutions

{(a,1),(6,1)}} {(a,2),(b,2)}, {(a,3), (6,3)), {(a,1), (b,2)};

{(a,1), (6,3), (a,2), (b,1)}, {(a,2), (b,3),}, {(a,3), (b,1,)}, {(2,3),(6,2}}, ring functions in ally

-Aftenshaft

Inferior Umbilicus

A Typical Wing Feather

Barb

·Barb

•Trough"

barbule

-The Tower

barbule

with hooks

A Simplified diagram to Show Interlocking Bar bules

Describe the structures of a bara's egg and give the function perfformed by each part the you mention. (b) Write a short aco

A hemed bird;

of the re-history

(a) The egg deposited by a female bird, 16

enclosed by a hard covering, the eggshell. The shell, made of che lk, is porous, allowing gases to pass through it. The contents, enclosed in a shell membrane, consist of a store of thin food-solution, the egg albumen, and a slightly thicker coloured yolk which contains a high proportion of fat resting on the upper surface of the yolk is the living part of the egg or called the germinal disc. The yolk is surrounded by the lbumen. Through the albumen are two twist cords, each running from the yolk to the shell membrane, thus the yolk is held in position. During the development of the embry both the yolk and the albumen serve as the parts for food supplying, while the germina. diad develop into the embryo.

Air Chamber

Germinal

disc

Egg Shell -Shell membrane

Albumen

A cord holds, the yolk Yolk

LS OF A Bird's Egg

The life-history of sparrow is described as the follows

During mating season, the male bird singa all the time. The femle bird constructs their] nest in a tree or close to humen houses,

The eggs are fertilized in the body of the female bird and are deposited in the in nest. Incubation takes about 12 days. After hatching the young birds are reared by the le. About a week latter, the young mag leave their nest and follow the male shout ing for food, while the mother bird again selects a suitable place to bui id a second nest.. Fins lly, the young leave their parent and live independent Ly..

Questions for this week:

1. Describe, with the aid of s fully labell96

diagram, the external features of a frog, What mentioned features of a frog do you consider as the adaptation of the frog to its environment? 2. Jut line the changes that occur in the development

of a newly formed tadpole to an adult frog. Indicate clearly how (1) feeding and (ii) breathing are accomplished at each stage of the development.

(4) Suppose g : X→Y, Which of following: 13 always

true: (a) f(x) = Y; (b) f(K) SY

(c) I(A) CY

(5) Let f D→→→→ D and g • D→→→D be defined by

f(x) = x2 - 2, g(x) • 3x + 1. Find conditions which define the composite functions g(f),

x2 - 2) + 1 = 3x2-

f(g) + (3x + 1)2 - 2(3x + 1)

(8)-3(3x + 1)

fi) - (2 -2x)?

252 LX

12. Exercises for the wee

week,

(1) Given F:DD and DD, whare fix),

= x2 - 3x + 1, and g(x). -5x-4, find f(g) and gig).,

Given A - 21,2,3,4) and B - 21,4,9,16, 13)18), If A→B is defined by f(x) = x2; state whether f is one-one or onto for one-ons and onto.

(3) Given A={1,2,3,4,5) and B11,2,4,6,7,8,9)

If AB is defined by f(x) = 3, state whether fis onto or one-one.

Stete

(4) Let f :D→ D be defined by f(x) whether is one-one and cnto. -(5) lat f:D→→ D be defined by f(x)

Find 1(2)−1(-3); £{(2)),Te