11th maths EXERCISE 1.3

JK MATHS  11th Maths Solutions 

Chapter 1 Sets Ex 1.3

Question 1.
Suppose that 120 students are studying in 4 sections of eleventh standard in a school. Let A denote the set of students and B denote the set of the sections. Define a relation from A to B as “x related toy if the student x belongs to the section y”. Is this relation a function? What can you say about the inverse relation? Explain your answer.
Solution:
(i) A = {set of students in 11th standard}
B = {set of sections in 11sup>th standard}
R : A ➝ B ⇒ x related to y
⇒ Every students in eleventh Standard must in one section of the eleventh standard.
⇒ It is a function.
Inverse relation cannot be a function since every section of eleventh standard cannot be related to one student in eleventh standard.

Question 2.
Write the values of f at – 4, 1, -2, 7, 0 if

Solution:

f(-4) = -(-4) + 4 = 8

f(1) = 1 – 12 = 0

f(-2) = (-2)2 – (-2) = 4 + 2 = 6

f(7) = 0

f(0) = 0

Question 3.

Write the values of f at -3, 5, 2, -1, 0 if

Solution:

f(-3) = (-3)2 – 3 – 5 = 9 – 8 = 1

f(5) = (5)2 + 3(5) – 2 = 25 + 15 – 2 = 38

f(2) = 4 – 3 = 1

f(-1) = (-1)2 + (-1) – 5 = 1 – 6 = -5

f(0) = 0 – 3 = -3

Samacheer Kalvi 11th Maths Solutions Chapter 1 Sets Ex 1.3

Question 4.

State whether the following relations are functions or not. If it is a function check for one-to-oneness and ontoness. If it is not a function, state why?

(i) If A = {a, b, c] and/= {(a, c), (b, c), (c, b)};(f: A ➝ A).

(ii) If X = {x, y, z} and/= {(x, y), (x, z), (z, x)}; (f: X ➝ X).

Solution: 

(i) f : A ➝ A

It is a function but it is not 1 – 1 and not onto function. 

(ii) f : X ➝ X

x ∈ X (Domain) has two images in the co-domain x. It is not a function.

Question 5.

Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Give a function from A ➝ B for each of the following:

(i) neither one-to-one nor onto.

(ii) not one-to-one but onto.

(iii) one-to-one but not onto.

(iv) one-to-one and onto.

Solution:

A = {1, 2, 3, 4}

B = {a, b, c, d}.

R = {(1, b) (2, b) (3, c) (4, d)} is not 1-1 and not onto

(iii) Not possible

(iv)

Question 6.

Find the domain of 11−2sinx

Solution:

Question 7.

Find the largest possible domain of the real valued function f(x) = 4−x2√x2−9√

Solution:

∴ No largest possible domain

The domain is null set

Question 8.

Find the range of the function 12cosx−1

Solution:

The range of cos x is – 1 to 1

Question 9.
Show that the relation xy = -2 is a function for a suitable domain. Find the domain and the range of the function.
Solution:
(i) Let f: R → R defined as f: x → −2x then
f(x) = −2x or y = −2x
⇒ xy = – 2
f (x) is not a function since f(x) is not defined for x = 0

(ii) Let f: R – {0} → R defined as f(x) = −2x
⇒ y = −2x = xy = – 2
f is one – one but not onto because 0 has no preimage.
f : R – {0} → R {0} is a function which is one- one and onto
Domain = R – {0}
Range = R – {0}

Question 10.
If f, g : R ➝ R are defined by f(x) = |x| + x and g(x) = |x| – x, find gof and fog.
Solution:

Question 11.

If f, g, h are real-valued functions defined on R, then prove that

(f + g)oh = foh + goh. What can you say about fo(g + h)? Justify your answer.

Solution:

Let f + g = k

= (f + g((h(x))

= f[h(x)] + g [h(x)]

= foh + goh

(i.e.,)(f + g)(o)h = foh + goh

fo(g + h) is also a function

 

Question 12.

If f: R ➝ R is defined by f(x) = 3x – 5, prove that f is a bijection and find its inverse.

Solution:

P(x) = 3x – 5

Let y = 3x – 5 ⇒ 3x = y + 5

Question 13.

The weight of the muscles of a man is a function of his bodyweight x and can be expressed as W(x) = 0.35x. Determine the domain of this function.

Solution:

Given W(x) = 0.35x

W(0) = W(1) = 0.35, W(2) = 0.7 ………….. W ( ∞ ) = ∞

Since x. denotes the bodyweight of a man, it will take only positive integers. That is x > 0.

W(x) : (0, ∞) → (0, ∞)

Domain = (0, ∞) , Range = (0, ∞)

Question 14.

The distance of an object falling is a function of time t and can be expressed as s(t) = -16t2. Graph the function and determine if it is one-to-one.

Solution:

s(t) = -16t2

Suppose S(t1) = S(t2)

since time cannot be negative, we to take t1 = t2

Hence it is one-one.

t 0 1 2 3

s 0 -16 -64 -144

Question 15.

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:

Given the cost of airfare function and fuel surcharge functions are as follows.

C(m) = 0.4 m+ 50 ———- (1)

S (m) = 0.03 m ———- (2)

Total cost of a ticket = C(m) + S(m)

f(x) = 0.4 m + 50 + 0.03 m

f(x) = 0.43 m + 50

Given m = 1600 miles

The cost of Airfare for flying 1600 miles

f( 1600 ) = 0.43 × 1600 + 50

= 688 + 50

= 738

∴ Airfare for flying 1600 miles is Rs. 738.

Question 16.

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(x) = 30, 000 + 0.04x, where x is merchandise rupee value

S(x) = 25000 + 0.05 x

(A + S) (x) = A(x) + S(x)

= 30000 + 0.04x + 25000 + 0.05 x

= 55000 + 0.09x

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

They each sell x = 1,50,00,000 worth of merchandise

(A + S) x = 55000 + 0.09 (1,50,00,000)

= 55000 + 13,50,000

∴ Total income of family = ₹ 14,05,000

Question 17.

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 the Indian rupee.

Solution:

f(x) = 1. 23x where x is number of American dollars.

g(y) = 50.50y where y is number of Singapore dollars.

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

= g(1. 23x)

= 50.50 (1.23x)

= 62.115 x

Question 18.

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:

Number of customers = 200 – x

Cost of one meal = Rs. 100

Cost of (200 – x) meals = (200 – x) × 100

Menu price of the meal = Rs. x

∴ Total menu price of (200 – x) meals = (200 – x) x

Profit = Menu price – Cost

= (200 – x) x – (200 – x) 100

Profit = (200 – x) (x – 100)

Question 19.

The formula for converting from Fahrenheit to Celsius temperatures is y=5x9−1609

Find the inverse of this function and determine whether the inverse is also a function.

Solution:

Question 20.
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