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**Ques 1: ****The relation f is defined by **

It is observed that for

0 ≤ *x* < 3, *f*(*x*) = *x*^{2}

3 < *x* ≤ 10, *f*(*x*) = 3*x*

Also, at *x* = 3, *f*(*x*) = 3^{2} = 9 or *f*(*x*) = 3 × 3 = 9

i.e., at *x* = 3, *f*(*x*) = 9

Therefore, for 0 ≤ *x* ≤ 10, the images of *f*(*x*) are unique.

Thus, the given relation is a function.

The relation *g* is defined as

It can be observed that for *x* = 2, *g*(*x*) = 2^{2} = 4 and *g*(*x*) = 3 × 2 = 6

Hence, element 2 of the domain of the relation *g* corresponds to two different images i.e., 4 and 6. Hence, this relation is not a function.

**Ques 2: ****If f(x) = x^{2}, find **

**Ques 3: ****Find the domain of the function ****Ans: **The given function is .

It can be seen that function *f* is defined for all real numbers except at *x* = 6 and *x* = 2.

Hence, the domain of *f* is **R** – {2, 6}.

**Ques 4: ****Find the domain and the range of the real function f defined by **

It can be seen that is defined for (

i.e., is defined for

Therefore, the domain of

As

Therefore, the range of

**Ques 5: ****Find the domain and the range of the real function f defined by f (x) = |x – 1|.**

It is clear that |

∴Domain of

Also, for

Hence, the range of

**Ques 6: ****Let ****be a function from R into R. Determine the range of f.**

The range of *f* is the set of all second elements. It can be observed that all these elements are greater than or equal to 0 but less than 1.

[Denominator is greater numerator]

Thus, range of *f* = [0, 1)

**Ques 7: Let f, g: R → R be defined, respectively by f(x) = x +1, g(x) = 2x – 3. Find f + g, f – g and f/g**

(*f* + *g*) (*x*) = *f*(*x*) + *g*(*x*) = (*x* +1) (2*x* – 3) = 3*x* – 2

∴(*f+ g*) (*x*) = 3*x* – 2

(*f – g*) (*x*) = *f*(*x*) – *g*(*x*) = (*x* + 1) – (2*x* – 3) = *x* + 1 – 2*x* + 3 = – *x* + 4

∴ (*f – g*) (*x*) = –*x* + 4

**Ques 8: ****Let f = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.**

*f*(*x*) = *ax* + *b*

(1, 1) ∈ *f*

⇒ *f*(1) = 1

⇒ *a* × 1 + *b* = 1

⇒ *a* + *b* = 1

(0, –1) ∈ *f*

⇒ *f*(0) = –1

⇒ *a* × 0 + *b* = –1

⇒ *b* = –1

On substituting *b* = –1 in *a* + *b* = 1, we obtain *a* (–1) = 1 ⇒ *a* = 1 + 1 = 2.

Thus, the respective values of *a* and *b* are 2 and –1.

**Ques 9: ****Let R be a relation from N to N defined by R = {( a, b): a, b ∈ N and a = b^{2}}. Are the following true?**

(i) It can be seen that 2 ∈ **N**;however, 2 ≠ 2^{2} = 4.

Therefore, the statement “(*a*, *a*) ∈ R, for all *a* ∈ **N**” is not true.

(ii) It can be seen that (9, 3) ∈ **N** because 9, 3 ∈ **N** and 9 = 3^{2}.

Now, 3 ≠ 9^{2} = 81; therefore, (3, 9) ∉ **N**

Therefore, the statement “(*a*, *b*) ∈ R, implies (*b*, *a*) ∈ R” is not true.

(iii) It can be seen that (9, 3) ∈ R, (16, 4) ∈ R because 9, 3, 16, 4 ∈ **N** and 9 = 3^{2} and 16 = 4^{2}.

Now, 9 ≠ 4^{2} = 16; therefore, (9, 4) ∉ **N**

Therefore, the statement “(*a*, *b*) ∈ R, (*b*, *c*) ∈ R implies (*a*, *c*) ∈ R” is not true.

**Ques 10: ****Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?**

∴A × B = {(1, 1), (1, 5), (1, 9), (1, 11), (1, 15), (1, 16), (2, 1), (2, 5), (2, 9), (2, 11), (2, 15), (2, 16), (3, 1), (3, 5), (3, 9), (3, 11), (3, 15), (3, 16), (4, 1), (4, 5), (4, 9), (4, 11), (4, 15), (4, 16)}

It is given that *f* = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

(i) A relation from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B.

It is observed that *f* is a subset of A × B.

Thus, *f* is a relation from A to B.

(ii) Since the same first element i.e., 2 corresponds to two different images i.e., 9 and 11, relation *f* is not a function.

**Ques 11: ****Let f be the subset of Z × Z defined by f = {(ab, a + b): a, b ∈ Z}. Is f a function from Z to Z: justify your answer.**

We know that a relation

Since 2, 6, –2, –6 ∈

i.e., (12, 8), (12, –8) ∈

It can be seen that the same first element i.e., 12 corresponds to two different images i.e., 8 and –8. Thus, relation

**Ques 12: ****Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find the range of f.**

Prime factor of 9 = 3

Prime factors of 10 = 2, 5

Prime factor of 11 = 11

Prime factors of 12 = 2, 3

Prime factor of 13 = 13

∴

The range of

∴Range of

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