Question 1: Find and in each of the following:

i) and ii) and

Answer:

i) Given and

is given by

is given by

is given by

is given by

is given by

is given by

ii) Given and

is given by

is given by

is given by

is given by

is given by

is given by

Question 2: Let and . Describe:

i) ii) iii) iv) . Find the domain in each case.

Answer:

We observe is defined for all . Hence Domain

Also is defined for all . Hence Domain

Therefore Domain Domain

i) is given by

Hence Domain

ii) is given by

Hence Domain

iii) is given by

Hence Domain

iv) is given by

Now for for all except

Hence Domain

Question 3: If be defined on and is given by

and . Find .

Answer:

Given

Now for

Question 4: Let and be two real functions defined by and . Then, describe each of the following functions:

i) ii) iii) iv) v) vi) vii) viii)

Answer:

Given and

We observe that is defined for

Hence Domain

Similarly, is defined for

Hence Domain

Now Domain Domain

i) is given by

ii) is given by

iii) is given by

iv) is given by

v) is given by

vi) is defined by

vii) is given by

viii) will be real for all values of except for and

Therefore is defined by

Question 5: If and , then determine each of the following functions:

i) ii) iii) iv) v)

Also find v)

Answer:

Given and

For is defined for all

Therefore Domain

For is defined for all

Therefore Domain

Therefore Domain Domain

i) is given by

ii) is given by

iii) Domain

will be real for all except

Therefore Domain Domain Domain

is defined as

iv)

is real if

and

and

Domain Domain Domain

is defined by

v)

does not exist because

Question 6: If are real functions defined by and , then find the values of and .

Answer:

Given and

will be real for all Domain

will be real for all except Domain

will be real for all Domain

Domain Domain Domain

is given by

does not exist as does not lie in domain

Question 7: The function f is defined by

Draw the graph of

Answer:

Given

Question 8: Let be defined respectively by . Find , and .

Answer:

Given

will be real for all Domain

) will be real for all Domain

Domain Domain

is defined by

is defined by

is real for

is give by

Question 9: Let and be defined by and . Find and .

Answer:

Given and

and

Domain Domain

is defined as

is defined as

is defined as

Question 10: Let and be two real functions. Find and

Answer:

Given and

will be real for all Domain

Similarly, will be real for all Domain

is defined as

is defined as

is defined as

is defined as