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f(x) = \(\sqrt{x-3} + \frac {1}{log(5-x)}\)

For f to be defined,

x – 3 ≥ 0, 5 – x > 0 and 5 – x ≠ 1 

x ≥ 3, x < 5 and x ≠ 4 

∴ Domain of f = [3, 4) ∪ (4, 5)

We have to prove that, 2/5 < log₁₀ 3 < 1/2

i.e., to prove that 2/5 < log₁₀ 3 and log₁₀ 3< 1/2

i.e., to prove that 2 < 5 log₁₀ 3 and 2 log₁₀ 3 < 1

i.e., to prove that 2 log₁₀ 10 < 5 log₁₀ 3 and 2 log₁₀ 10 ……

[∵ log_a a = 1]

i.e., to prove that log₁₀ 10² < log₁₀ 3⁵ and log₁₀ 3² < log₁₀ 10 

i.e., to prove that 10² < 3⁵ and 3² < 10 

i.e., to prove that 100 < 243 and 9 < 10 which is true

∴ 2/5 < log₁₀ 3 < 1/2

f = {(1, 3), (2, 4), (3, 5), (4, 6)}

g = {(3, 6), (4, 8), (5, 10), (6, 12)}

∴ f(1) = 3, g(3) = 6

f(2) = 4, g(4) = 8

f(3) = 5, g(5)=10

f(4) = 6, g(6) = 12 

gof) (x) = g (f(x))

(gof)(1) = g(f(1)) = g(3) = 6

(gof)(2) – g(f(2)) = g(4) = 8

(gof)(3) = g(f(3)) = g(5) = 10

(gof)(4) = g(f(4)) = g(6) = 12

∴ gof = {(1, 6), (2, 8), (3, 10), (4, 12)}

Given,

y = f(x) = (ax-b)/(bx-a)

⇒ f(y) = \(\frac{ay-b}{by-a}\) 

We need to prove that x = f(y).

We have,

y = \(\frac{ax-b}{bx-a}\)

⇒ y(bx – a) = ax – b 

⇒ bxy – ay = ax – b 

⇒ bxy – ax = ay – b 

⇒ x(by – a) = ay – b

⇒ x = \(\frac{ay-b}{by-a}\) = f(y)

∴ x = f(y) 

Thus, 

x = f(y).

f(x) = ax² + bx + 2

f(1) = 3

∴ a(1)² + b(1) + 2 = 3

∴ a + b = 1 ….(i)

f(4) = 42

∴ a(4)² + b(4) + 2 = 42

∴ 16a² + 4b = 40

Dividing by 4, we get

4a + b = 10 …..(ii)

Solving (i) and (ii), we get

a = 3, b = -2

To prove: (f o g) = I_R = (g o f).

Formula used: (i) f o g = f(g(x))

(ii) g o f = g(f(x))

Given: (i) f : R → R : f(x) = (2x - 3)

(ii) \(g:R→R:g(x)=\frac{1}{2}(x+3)\)

Solution: We have,

f o g = f(g(x))

\(=f(\frac{1}{2}(x+3))\)

\(=[2(\frac{1}{2}(x+3))-3]\)

= x + 3 – 3

= x

= I_R

g o f = g(f(x))

= g(2x - 3)

\(=\frac{1}{2}(2x-3+3)\)

\(=\frac{1}{2}(2x)\)

= x

= I_R

Clearly we can see that (f o g) = IR = (g o f) = x

Hence Proved.

Description : The method sqrt( ) returns the square root of x for x > 0. 

Syntax: Following is the syntax for sqrt() method 

import math 

math.sqrt(x) 

Note: This function is not accessible directly so we need to import math module and then we need to call this function using math static object. 

Parameters: x —This is a numeric expression. 

Return Value : This method returns square root of x for x > 0. 

Example: The following example shows the usage of sqrt() method.

# !/usr/bin/python 

import math # This will import math module print “math.sqrt(100): “, 

math.sqrt(100) 

print “math.sqrt(7): “, 

math.sqrt(7) 

print “math.sqrt(math.pi): “, math.sqrt(math.pi)

This will produce the following result:

math.sqrt(100): 10.0 

math.sqrt(7): 2.64575131106

math.sqrt(math.pi): 1.77245385091

Correct option is (d) More than three times

Correct option is (a) Library functions

Description: The method cos() returns the cosine of x radi Answer: 

Syntax : Following is the syntax for cos() method 

cos(x)

Note: This function is not accessible directly so we need to import math module and then we need to call this function using math static object.

Parameters : x—This must be a numeric value. 

Return Value : This method returns a numeric value between-1 and 1 which represents the cosine of the angle. 

Example: The following example shows the usage of cos() method

# !/usr/bin/python 

import math 

print “cos(3):”, 

math.cos(3) 

print “cos(-3):”, 

math.cos(-3) 

print “cos(0):”, 

math.cos(0) 

print “cos(math.pi):”, 

math.cos(math.pi) 

print “cos(2*math.pi):”, 

math.cos(2*math.pi)

This will produce the following result:

cos(3) >0.9899924966 

cos(-3) >0.9899924966 

cos(0): 1.0 

cos(math.pi) >1.0

Return Statement: The keyword return is used to terminate function and return a value to its caller. The return statement may also be used to exit a function without returning a value. The return statement may or may not include an expression.
Example Program:

# include<stdio.h>
void main (void)
{
float max (float, float, float);
float a, b, c maximum;
printf (“Enter three numbers\n”);
scanf (“%d %d %d”, &a, &b &c);
maximum = max (a, b, c);
printf(“maximum = %d, maximum);
}
float max (float x, float y, float z)
}
if (x > y)
{
if (x > z)
return (x);
}
else
{
if (y > z)
return (z);
else
return (z);
}
}

Output:
Enter three numbers
2 4 8
maximum = 8

True

 True

True

False.

Returns e**x.

# tic-tac-toe program 

import random def drawboard (board): 

# This function is used to print the board

print (‘ | | ‘) 

print (‘ ‘ + board [7] + ‘ j ‘ + board [8] + ‘ | ‘ + board [9]) print (‘| | ‘) 

print (‘ ‘) 

print (‘| | ‘) 

print (‘ ‘ + board [4] + ‘ | ‘ + board [5] + ‘ | ‘ + board [6]) print (‘| | ‘) 

print (‘ ‘) 

Print (‘ | | ‘) 

print (‘ ‘ + board [1] + ‘ | ‘ + board [2] + ‘ | ‘ + board [3]) print (‘| | ‘) 

def input player letter (): 

# This function is used to display the user typed letter Letter = ” 

while not (letter = = ‘X’ or letter = = ‘O’): 

print (‘Do you want to be X or O ?’) 

letter = input () .upper () 

if letter = ‘X’ return [‘X’, ‘O’] 

else: 

return [‘O’, ‘X’] def whogoesFirst (): 

# This function is used to randomly choose the player 

if randomrandint (0,1) = = 0 : return ‘Computer’ 

else:

return ‘Player’ def Play Again (): 

# This function is used when player wants play again print 

(‘Do you want to play again ? (Yes or no)’) return input (), lower ().startswith (‘y’) 

def makeMove (board, letter, move): board[move] = letter def is Winner (bo, le): 

# This function returns true when plays has won return

(60[7] = = le and 60[8] = = le and 60[9] = = le) or 

(60[4] = = le and 60[5] = = le and 60[6] = = le) or 

(60[1 ] = = le and 60[2] = = le and 60[3] = = le) or 

(60[7] = = le and 60[4] = = le and 60[1] = = le) or 

(60[8] = = le and 60[5] = = le and 60[2] = = le) or 

(60[9] = = le and 60[6] = = le and 60[3] = = le) or 

(60[7] = = le and 60[5] = = le and 60[3] = = le) or 

(60[9] = = le and 60[5] = = le and 60[1] = = le) or

def getBoardcopy(board): 

# This function make a duplicate of the board dupeBoard = [ ] 

for i in board : 

dupeBoard.append(i) 

return depeBoard 

def isSpaceFree (board, move): 

# This function returns Ttue if the passed

move is free return boardfmove] = = ‘ ‘ def getPlayerMove(board): 

# This function is used to take player’s move move = ‘ ‘ 

while move not in’12345678 9’.split () or 

not is spaceFree (board, int(move)): 

print (‘What is your next move ? (1-9)’) 

move = input () return int(move) 

defchooseRandomMoveFromList(board, movelist): 

# This function returns a valid move from the passed list possible Moves = [ ] for i in moveList: 

if is SpaceFree (board, i):

possibleMOves.append(i)

if len(possibleMoves)! = 0 : 

return random.choice (possibleMoves)

else: 

return – None 

defgetComputerMove(board, computerletter): 

# This function returns the computer’s letter

if computer letter = = ‘X’: 

player letter = ‘O’ 

else: 

player letter = ‘X’ 

for i in range (1,10): 

copy = getBoardCopy (board) 

if isSpaceFree (copy, i): 

makeMove (copy, computerletter, i) 

if is Winner (copy, computerletter): 

return i for i in range (1,10): 

copy = getBoard Copy (board) 

if isSpaceFree (copy, i): 

makeMove (copy, player letter, i) 

if is Winner (copy, player letter): 

return i 

move=chooseRandomMoveFromList (board, [1, 3, 7, 9]) 

if move ! = None : 

return move 

if isSpaceFree (board, 5): 

return 5 

return ChooseRandomMoveFromList (board, [2,4,6,8]) 

def is BoardFull (board): 

# This function returns true if all space on board has been taken for i in range (1,10): 

if isSpaceFree (board, i): 

return False 

return True print 

(“Welcome to Tic-Tac-Toe !”) 

while True: 

the Board = [” * 10

playerletter, computerletter = input

 playerletter() 

turn = whoGoesFirst() 

print (‘The’ + turn + ‘will go first.’)

 gamelsPlaying = True 

while gemelsPlaying: 

if turn = = ‘player’: 

drawBoard (theBoard) 

move = getPlayerMove (theBoard)

makeMove (theBoard, playerletter, move) 

if is Winner (theBoard, playletter):

drawBoard (theBoard) 

print (‘Hooray ! You have won !’)

gamelsPlaying = False 

esle: 

if is BoardFull (theBoard): 

drawBoard (theBoard) 

print (‘The game is a tie.”) 

break 

else: 

turn = ‘computer’ 

else: 

move=getCompterMove(theBoard, Computerletter) 

makeMove (theBoard, computer letter, move) 

if is Winner (theBoard, computerletter): drawBoard (theBoard) 

print(‘The computer has beaten you!’) gamelsPlaying = False 

else: 

if isBoardFull (theBoard): 

drawBoard (theBoard) 

print (‘The game is a tie !’) 

break 

else: 

turn = ‘player’ if not play Again (): 

break

Description : The method exp( ) returns exponential of x : ex. 

Example: 

Syntax: Following is the syntax for exp() method 

import math 

math.exp(x) 

Note: This function is not accessible directly so w e need to import math module and then we need to call this function using math static object. 

Parameters : x—This is a numeric expression. Return Value: This method returns exponential of x: ex. 

Example: The following example shows the usage of exp() method 

# !/usr/bin/py thon 

import math # This will import math module

 print “math.exp(-45.17): “, 

math.exp(-45.17) 

print “math.exp(100.12):”, 

math.exp(100.12) 

print “math.exp(100.72):”, 

math.exp(100.72) 

print “math.exp(119L):”, 

math.exp(119L) 

print “math.exp(math.pi):”, math.exp(math.pi), 

This will produce the following result

math.exp(-45.17):2.41500621326e-20

math.exp(100.12):3.03084361407e+43

math.exp(100.72): 5.52255713025e+43

math.exp(119L):4.7978133273e+51

 math.exp(math.pi): 23.1406926328

Merits of modular programming 

• It reduces the size of the program 
• Less chance of error occurrence 
• Reduces programming complexity 
• Improves reusability

The length(number of characters) is 12 including space.

n = 0 is the value of n.

(a) call by value

The process of converting big programs into smaller programs is known as modularisation. This small programs are called modules or sub programs or functions. C++ supports modularity in programming called functions.

1. K is a local variable in the function initialize() – It is not accessible in main()

2. Making the variable K as global we can correct the error.

(d) cout.write(wond, 10);

char str[20] = “Computer”;

OR

char str[20]; strcpy(str,”Computer”); (The header le should be included).

In the case of a sub function a return statement helps to terminate the sub function and return back to the main function or called function. But in the case of a main function it terminates the program.