## EXERCISE 2.3

#### Page No 44:

#### Question 1:

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(iii) {(1, 3), (1, 5), (2, 5)}

#### Answer:

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

Since 2, 5, 8, 11, 14, and 17 are the elements of the domain of the given relation having their unique images, this relation is a function.

Here, domain = {2, 5, 8, 11, 14, 17} and range = {1}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

Since 2, 4, 6, 8, 10, 12, and 14 are the elements of the domain of the given relation having their unique images, this relation is a function.

Here, domain = {2, 4, 6, 8, 10, 12, 14} and range = {1, 2, 3, 4, 5, 6, 7}

(iii) {(1, 3), (1, 5), (2, 5)}

Since the same first element i.e., 1 corresponds to two different images i.e., 3 and 5, this relation is not a function.

#### Question 2:

Find the domain and range of the following real function:

(i) *f*(*x*) = –|*x*| (ii)

#### Answer:

(i) *f*(*x*) = –|*x*|, *x* ∈ R

We know that |*x*| =

Since *f*(*x*) is defined for *x* ∈ **R**, the domain of *f* is **R.**

It can be observed that the range of *f*(*x*) = –|*x*| is all real numbers except positive real numbers.

∴The range of *f* is (–, 0].

(ii)

Sinceis defined for all real numbers that are greater than or equal to –3 and less than or equal to 3, the domain of *f*(*x*) is {*x* : –3 ≤ *x* ≤ 3} or [–3, 3].

For any value of *x* such that –3 ≤ *x* ≤ 3, the value of *f*(*x*) will lie between 0 and 3.

∴The range of *f*(*x*) is {*x*: 0 ≤ *x* ≤ 3} or [0, 3].

#### Question 3:

A function *f* is defined by *f*(*x*) = 2*x* – 5. Write down the values of

(i) *f*(0), (ii) *f*(7), (iii) *f*(–3)

#### Answer:

The given function is *f*(*x*) = 2*x* – 5.

Therefore,

(i) *f*(0) = 2 × 0 – 5 = 0 – 5 = –5

(ii) *f*(7) = 2 × 7 – 5 = 14 – 5 = 9

(iii) *f*(–3) = 2 × (–3) – 5 = – 6 – 5 = –11

#### Question 4:

The function ‘*t*’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by.

Find (i) *t* (0) (ii) *t* (28) (iii) *t* (–10) (iv) The value of C, when *t*(C) = 212

#### Answer:

The given function is.

Therefore,

(i)

(ii)

(iii)

(iv) It is given that *t*(C) = 212

Thus, the value of *t*, when *t*(C) = 212, is 100.

#### Question 5:

Find the range of each of the following functions.

(i) *f*(*x*) = 2 – 3*x*, *x* ∈ **R**, *x* > 0.

(ii) *f*(*x*) = *x*^{2} + 2, *x*, is a real number.

(iii) *f*(*x*) = *x*, *x* is a real number

#### Answer:

(i) *f*(*x*) = 2 – 3*x*, *x* ∈ **R**, *x* > 0

The values of *f*(*x*) for various values of real numbers *x* > 0 can be written in the tabular form as

| 0.01 | 0.1 | 0.9 | 1 | 2 | 2.5 | 4 | 5 | … |

| 1.97 | 1.7 | –0.7 | –1 | –4 | –5.5 | –10 | –13 | … |

Thus, it can be clearly observed that the range of *f* is the set of all real numbers less than 2.

i.e., range of *f* = (–, 2)

**Alter:**

Let *x* > 0

⇒ 3*x* > 0

⇒ 2 –3*x* < 2

⇒ *f*(*x*) < 2

∴Range of *f* = (–, 2)

(ii) *f*(*x*) = *x*^{2} + 2, *x*, is a real number

The values of *f*(*x*) for various values of real numbers *x* can be written in the tabular form as

| 0 | ±0.3 | ±0.8 | ±1 | ±2 | ±3 | … | |

| 2 | 2.09 | 2.64 | 3 | 6 | 11 | ….. |

Thus, it can be clearly observed that the range of *f* is the set of all real numbers greater than 2.

i.e., range of *f* = [2,)

**Alter:**

Let *x* be any real number.

Accordingly,

*x*^{2} ≥ 0

⇒ *x*^{2} + 2 ≥ 0 + 2

⇒ *x*^{2} + 2 ≥ 2

⇒ *f*(*x*) ≥ 2

∴ Range of *f* = [2,)

(iii) *f*(*x*) = *x, x* is a real number

It is clear that the range of *f* is the set of all real numbers.

∴ Range of *f* = **R**