WORD PROBLEMS ON RELATIONS AND FUNCTIONS

Problem 1 :

The total cost of airfare on a given route is comprised of the base cost C and the fuel surcharge S in rupee. Both C and S are functions of the mileage m; C(m) = 0.4m + 50 and S(m) = 0.03m. Determine a function for the total cost of a ticket in terms of the mileage and find the airfare for flying 1600 miles.

Solution :

Total cost   =  C(m) + S(m)

  =  0.4m + 50 + 0.03m

T(m)  =  0.43 m + 50

To find the airfare for flying 1600 miles, we have to apply 1600 instead of m.

T(m)  =  0.43 (1600) + 50

  =  688 + 50

T(m)  =  738

So, the total cost of airfare for flying 1600 miles is 738.

Problem 2 :

A salesperson whose annual earnings can be represented by the function A(x) = 30, 000 + 0.04x, where x is the rupee value of the merchandise he sells. His son is also in sales and his earnings are represented by the function S(x) = 25, 000 + 0.05x. Find (A + S)(x) and determine the total family income if they each sell Rupees 1, 50, 00, 000 worth of merchandise.

Solution :

(A + S)(x)   =  A(x) + S(x)

A(x) = 30, 000 + 0.04x    -------(1)

S(x) = 25, 000 + 0.05x    -------(2)

(1) + (2)

   =  30, 000 + 0.04x + 25, 000 + 0.05x 

   A(x) + S(x)  =  55000 + 0.09x

Here x = 15000000

   A(x) + S(x)  =  55000 + 0.09(15000000)

  =  55000 +  1350000

Total income  =  1405000

So, the required income is 1405000.

Problem 3 :

The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day the function for exchanging Singapore Dollar to Indian Rupee is g(y) = 50.50y, where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee.

Solution :

The function for exchanging American dollars for Singapore Dollar :

f(x) = 1.23x

S.D  =  1.23 (A.D)

Here "x" stands for American dollar and f(x) stands for Singapore dollar.

A.D  = S.D/1.23  ----(1)

Exchanging Singapore Dollar to Indian Rupee is

g(y) = 50.50y

I.R  =  50.50 (S.D)

S.D  =  I.R/50.50

Applying S.D  =  I.R/50.50 in(1), we get

A.D  = (I.R/50.50)/1.23

A.D =  I.R/62.115

I.R  =  62.115 AD

So, the exchange rate of American dollars in terms of Indian rupee is I.R  =  62.115 AD.

Problem 4 :

The owner of a small restaurant can prepare a particular meal at a cost of Rupees 100. He estimates that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200−x. Express his day revenue, total cost and profit on this meal as functions of x.

Solution :

Cost price of the meal  =  100

Selling price  =  x 

Number of customers  =  200 - x

1 day revenue  =  No of customers ⋅ x

  =  (200 - x) ⋅ x

 1 day revenue  =  200 x - x²

Total cost  =  Cost of meal ⋅ No of customers

  =  100 ⋅ (200 - x)

  = 20000 - 100x 

Profit  =  Total cost - 1 day revenue

(200 - x) ⋅ x - 100 ⋅ (200 - x) 

Problem 5 :

The formula for converting from Fahrenheit to Celsius temperatures is y = (5x/9) − (160/9). Find the inverse of this function and determine whether the inverse is also a function

Solution :

 y = (5x - 160)/9

9y  = 5x - 160

5x  =  9y + 160

x  =  (1/5) (9y + 160)

f^-1 (x)  =  (1/5) (9x + 160)

f^-1 (x)  =  (9x/5) + (160/5)

f^-1 (x)  =  (9x/5) + 32

Inverse is also a function.

Problem 6 :

A simple cipher takes a number and codes it, using the function f(x) = 3x−4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x (by drawing the lines).

Solution :

f(x) = 3x−4

Let y = 3x - 4

3x  =  y + 4

x  =   (y + 4)/3

f^-1(x)  =  (x + 4)/3

Inverse is also a function.

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