__ RELATIONS AND FUNCTIONS- CLASS 10- CHAPTER 1-EXERCISE 1.3__

**Question 1.**

**Let f = {(x, y)|x, y ∈ N and y = 2x} be a relation on N. Find the domain, co-domain and range. Is this relation a function?**

**Solution:**

F = {(x, y)|x, y ∈ N and y = 2x}

x = {1, 2, 3,…}

y = {1 × 2, 2 × 2, 3 × 2, 4 × 2, 5 × 2 …}

R = {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10),…}

Domain of R = {1, 2, 3, 4,…},

Co-domain = {1, 2, 3…..}

Range of R = {2, 4, 6, 8, 10,…}

Yes, this relation is a function.

**Question 2.**

**Let X = {3, 4, 6, 8}. Determine whether the relation R = {(x, f(x))|x ∈ X, f(x) = x2 + 1} is a function from X to N ?**

**Solution:**

x = {3,4, 6, 8}

R = ((x, f(x))|x ∈ X, f(x) = X2 + 1}

f(x) = x2 + 1

f(3) = 32 + 1 = 10

f(4) = 42 + 1 = 17

f(6) = 62 + 1 = 37

f(8) = 82 + 1 = 65

R = {(3, 10), (4, 17), (6, 37), (8, 65)}

R = {(3, 10), (4, 17), (6, 37), (8, 65)}

Yes, R is a function from X to N.

**Question 3.**

**Given the function**

**f : x → x2 – 5x + 6, evaluate**

**(i) f(-1)**

**(ii) f(2 a)**

**(iii) f(2)**

**(iv) f(x – 1)**

**Solution:**

f(x) = x2 – 5x + 6

(i) f (-1) = (-1)2 – 5 (-1) + 6 = 1 + 5 + 6 = 12

(ii) f (2a) = (2a)2 – 5 (2a) + 6 = 4a2 – 10a + 6

(iii) f(2) = 22 – 5(2) + 6 = 4 – 10 + 6 = 0

(iv) f(x – 1) = (x – 1)2 – 5 (x – 1) + 6

= x2 – 2x + 1 – 5x + 5 + 6

= x2 – 7x + 12

**Question 4.**

**A graph representing the function f(x) is given in figure it is clear that f(9) = 2.**

**(i) Find the following values of the function**

**(a) f(0)**

**(b) f(7)**

**(c) f(2)**

**(d) f(10)**

**(ii) For what value of x is f (x) = 1?**

**(iii) Describe the following**

**(i) Domain**

**(ii) Range.**

**(iv) What is the image of 6 under f?**

**Solution:**

(a) f(0) = 9

(b) f(7) = 6

(c) f(2) = 6

(d) f(10) = 0

(ii) At x = 9.5, f(x) = 1

(iii) Domain = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

= {x |0 < x < 10, x ∈ R}

Range = {x|0 < x < 9, x ∈ R}

= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

(iv) The image of 6 under f is 5.

**Question 5.**

**Let f(x) = 2x + 5. If x ≠ 0 then find f(x+2)−f(2)x**

**Solution:**

Given f(x) = 2x + 5, x ≠ 0.

**Question 6.**

**A function is defined by f(x) = 2x – 3**

**(i) find f(0)+f(1)2**

**(ii) find x such that f(x) = 0.**

**(iii) find x such that f(x) = x.**

**(iv) find x such that f(x) = f(1 – x).**

**Solution:**

Given f(x) = 2x – 3

(i) find f(0)+f(1)2

f(0) = 2(0) – 3 = -3

f(1) = 2(1) – 3 = -1

∴ f(0)+f(1)2=−3−12=−42 = -2

(ii) f(x) = 0

⇒ 2x – 3 = 0

2x = 3

x = 32

(iii) f(x) = x

⇒ 2x – 3 = x ⇒ 2x – x = 3

x = 3

(iv) f(x) = f(1 – x)

2x – 3 = 2(1 – x) – 3

2x – 3 = 2x – 2x – 3

2x + 2x = 2 – 3 + 3

4x = 2

x = 24

x = 12

**Question 7.**

**An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal squares from the corners and turning up the sides as shown in figure. Express the volume V of the box as a function of x.**

**Solution:**

Volume of the box = Volume of the cuboid

= l × b × h cu. units

Here l = 24 – 2x

b = 24 – 2x

h = x

∴ V = (24 – 2x) (24 – 2x) × x

= (576 – 48x – 48x + 4×2)x

V = 4×3 – 96×2 + 576x

**Question 8.**

**A function f is defined by f(x) = 3 – 2x . Find x such that f(x2) = (f(x))2.**

**Solution:**

f(x) = 3 – 2x

f(x2) = 3 – 2×2

**Question 9.**

**A plane is flying at a speed of 500 km per hour. Express the distance d travelled by the plane as function of time r in hours.**

**Solution:**

Speed of the plane = 500 km/hr

Distance travelled in “t” hours

= 500 × t (distance = speed × time)

= 500 t

**Question 10.**

**The data in the adjacent table depicts the length of a woman’s forehand and her corresponding height. Based on this data, a student finds a relationship between the height (y) and the forehand length(x) as y = ax + b, where a, b are constants.**

**(i) Check if this relation is a function.**

**(iii) Find the height of a woman whose forehand length is 40 cm.**

**(iv) Find the length of forehand of a woman if her height is 53.3 inches.**

**Solution:**

(i) Given y = ax + b …………. (1)

The ordered pairs are R = {(35, 56) (45, 65) (50, 69.5) (55, 74)}

∴ Hence this relation is a function.

Substituting a = 0.9 in (2) we get

⇒ 65 = 45(.9) + b

⇒ 65 = 40.5 + b

⇒ b = 65 – 40.5

⇒ b = 24.5

∴ a = 0.9, b = 24.5

∴ y = 0.9x + 24.5

(iii) Given x = 40 , y = ?

∴ (4) → y = 0.9 (40) + 24.5

⇒ y = 36 + 24.5

⇒ y = 60.5 inches

(iv) Given y = 53.3 inches, x = ?

(4) → 53.3 = 0.9x + 24.5

⇒ 53.3 – 24.5 = 0.9x

⇒ 28.8 = 0.9x

⇒ x = 28.80.9 = 32 cm

∴ When y = 53.3 inches, x = 32 cm