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