Exercise 1.3: Functions

Exercise 1.3

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?

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 ?

3. Given the function f : x → x2 − 5x + 6 , evaluate

(i) f (-1)

(ii) f (2a)

(iii) f (2)

(iv) f (x − 1)

4. A  graph representing the function f (x) is  given in Fig.1.16 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 ?

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

6. A function f 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)

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 (Fig.1.17). Express the volume V of the box as a function of x.

8. A function f is defined by f (x) = 3 − 2x . Find x such that f (x 2 ) = (f (x))2

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 t in hours.

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.

(i) Find a and b.

(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.

Answers:

1. {1, 2, 3, 4,...} , {1,2,3,4}, {2, 4, 6, 8,...} , yes.

2. yes

3.(i) 12 (ii) 4a2 − 10a + 6 (iii) 0 (iv) x2 − 7x + 12

4.(i) (a) 9 (b) 6 (c) 6 (d) 0

(ii) 9.5 (iii) (a) {x / 0 ≤ x ≤ 10, x ∈ R} (b) {x / 0 ≤ x ≤ 9, x ∈ R} (iv) 5

5. 2

6.(i) -2 (ii) 3/2 (iii) 3 (iv) 1/2

7. 4x3 − 96x2 + 576x

8. 1

9. 500t

10.(i) Yes (ii) 0.9,24.5 (iii) 60.5 inches (iv) 32 cms