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**Miscellaneous Exercise**

**1. The relation **** is defined by ****. The relation **** is defined by ****. Show that **** is a function and **** is not a function.**

**Ans. **Given: and

At and

It is observed that takes unique value at each point in its domain [0, 10]. Therefore, is a function.

Now, and

At and

Therefore, does not have unique value at

Hence, is not a function.

**2. If **** find **

**Ans. **Given:

At

and

**3. Find the domain of the function **

**Ans. **Given:

is a rational function of

assumes real values of all except for those values of for which

Domain of function = R – {2, 6}

**4. Find the domain and range of the real function **** defined by **

**Ans. **Given: assumes real values if

Domain of

For

Range of = all real numbers 0 =

**5. Find the domain and range of the real function **** defined by **

**Ans. **Given:

The function is defined for all values of

Domain of = R

When ,

When ,

When ,

Range of = All real numbers 0 =

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

**Ans. **Here

Putting

Now, will be real if

Range of

**7. Let **** be defined respectively by **** Find **** and **

**Ans. **Given: and

Now,

And

And

**8. Let **** be a function from Z to Z defined by **** for some integers **** Determine **

**Ans. **Given: and

Now

……….(i)

And

……….(ii)

Solving eq. (i) and (ii), we get and

**9. Let R be a relation from N to N defined by R = **** Are the following true:**

**(i) **** R for all **** N**

**(ii) **** R implies **** R**

**(iii) **** R, **** R implies **** R**

**Ans. **Given: R =

**(i) **No, (3, 3) R because

**(ii) **No, (9, 3) R but (3, 9) R

**(iii) **No, (81, 9) R but (81, 3) R

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

**(i) **** is a relation from A to B.**

**(ii) **** is a function from A to B.**

**Justify your answer in each case.**

**Ans. (i)** Here A = {1, 2, 3, 4} and B = {1, 5, 9, 11, 15, 16}

= {(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)}

= {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

Now, (1, 5), (2, 9), (3, 1), (4, 5), (2, 11)

is a relation from A to B.

**11. Let **** be a subset of defined by **** Is **** a function from Z to Z? Justify your answer.**

**Ans. **We observed that = 4 and = 4

and

(4, 5) and (4, 4)

It shows that is not a function from Z to Z.

**12. Let A = {9, 10, 11, 12 13} and let **** be defined by **** the highest prime factor of **** Find the range of **

**Ans. **Here A = {9, 10, 11, 12, 13}

For

[and 3 is highest prime factor of 9]

For

[]

For

[]

For

[]

For

[]

Range of = {5, 11, 3, 13}

= {3, 5, 11, 13}