**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)}**

**Solution: **

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}Here, each element in domain is having unique/distinct image. So, the given relation is a function.

Domain = {2, 5, 8, 11, 14, 17}

Range of the function = {1}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}Here, each element in domain is having unique/distinct image. So, the given relation is a function.

Domain = {2, 4, 6, 8, 10, 12, 14}

Range of function = {1, 2, 3, 4, 5, 6, 7}

(iii) {(1, 3), (1, 5), (2, 5)}This relation is not a function since an element 1 corresponds to two elements/images i.e, 3 and 5.

Hence, this relation is not a function.

**Question 2.** **Find the domain and range of the following real function:**

**(i) f(x) = –|x| **

**(ii) f(x) = √(9 – x ^{2}) **

**Solution: **

(i)Given,f(x) = –|x|, x ∈ R

We know that, |x| =

x if x >= 0 -x if x < 0 Here f(x) = -x =

-x x >= 0 x x < 0 As f(x) is defined for x ∈ R, the domain of f is R.

It is also seen that the range of f(x) = –|x| is all real numbers except positive real numbers.

Therefore, the range of f is given by (–∞, 0].

(ii)f(x) = √(9 – x^{2})As √(9 – x

^{2}) is defined for all real numbers that are greater than or equal to –3 and less than or equal to 3, for 9 – x^{2}≥ 0.|x| <=3

So, the domain of f(x) is {x: –3 ≤ x ≤ 3} or Domain of f = [–3, 3].

For any value of x in the range [–3, 3], the value of f(x) will lie between 0 and 3.

Therefore, the range of f(x) is {x: 0 ≤ x ≤ 3} or we can say Range of f = [0, 3].

**Question 3. A function f is defined by f(x) = 2x – 5. Write down the values of**

**(i) f(0), (ii) f(7), (iii) f(–3)**

**Solution: **

Given, function, f(x) = 2x – 5.

(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 t(C) = 9C/5 + 32.**

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

Solution :

Here in ques , it is given that :

t(C) = 9C / 5 +32

So, (i) t(0) = 9(0) / 5 + 32

= 0 + 32

= 32

(ii) t(28) = 9(28) / 5 + 32

Taking LCM and solving ,

= ( 252 +160 ) / 5

= 412 / 5

(iii) t(-10) = 9(-10) / 5 + 32

= -18 + 32

= 14

(iv) Here , in this ques we have to find the value of C.

Given that , t(C) = 212,

9C / 5 + 32 = 212

9C / 5 = 180

9C = 180 X 5

C = 100

The value of C is 100.

**Question 5. Find the range of each of the following functions.**

**(i) f(x) = 2 – 3x, x ∈ R, x > 0.**

**(ii) f(x) = x2 + 2, x is a real number.**

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

**Solution:**

(i)Given f (x) = 2 – 3x, x ∈ R, x > 0∵ x > 0 ⇒ -3x < 0 (Multiplying both sides by -3)

⇒ 2 – 3x < 2 + 0 ⇒ f (x) < 2

∴ Hence, The range of f (x) is (-∞ , 2).

(ii)Given f (x) = x^{2}+ 2, x is a real numberWe know x

^{2}≥ 0 ⇒ x^{2}+ 2 ≥ 0 + 2⇒ x

^{2}+ 2 > 2 ∴ f (x) ≥ 2∴ Hence, The range of f (x) is [2, ∞).

(iii)Given f (x) = x, x is a real number.Let y = f (x) = x ⇒ y = x

∴ Range of f (x) = Domain of f (x)

∴ Hence, Range of f (x) is R. (f(x) takes all real values)