Unit1.1.2 : Function
- Author:
- AMBIK BARAL
Exercise 1.1.2
1. (a) Let such that and such that g(y) = z. Name the function defined from A to C.
Solution:
The function defined from A to C is .
1. (b) Write the difference between and .
Solution:
is the composite function of and and read as followed by .
is the composite function of and and read as followed by .
1. (c) Define composite function of and .
Solution:
If and be any two functions then the new function devinded from is called composite function of anf . It is denoted by .
2. (a) Given and are two real-valued functions defined as below. Find (i) domain of (ii) domain of (iii) domain of (iv) range of in each of the following if exist.
(a) and .
Solution:
(i) Domain of
(ii) Domain of
(iii) Since range of domain of so, , does not exists.
(iv) Range of = range of
(b) and
Solution:
(i) Domain of
(ii) Domain of
(iii) As range of is not a subset of domain of , so can not be defined. Therefore, does not exists.
(iv) Range of = Range of
3. Given that and , calculate:
(a) and
Solution:
Also,
(b)
Solution:
Also,
(c) and
Solution:
Also,
(d) and
Solution:
Also,
4. (a) If and then find and .
Solution:
Also,
4. (b) Given that and , find and . (Taking composition of two functions as a single function).
Solution:
5. If and are linear functions, what can you say about the domain of and ? Explain.
Solution:
The domain of = Doman of .
The domain of = Domain of .
6. Dolma determines the domain of by examining only the formula for Is her approach valid? Why or why not?
Solution:
If Dolma determines the domain of by examining only the formula of then her approach may not be valid. We know the domain of is same as the domain of If formula of contains radical sign or fraction having zero values in the denominator for some values of then the domain of cannot be examined just by the formula of
For example,
(i)Let’s define the functions an by and then which is defined at but is not defined at That means 0 is not the element in the domain of and hence 0 is not the element in the domain of .
(ii)Let’s define the functions an by and then which is defined at but is not defined at That means 3 is not the element in the domain of g and hence 3 is not the element in the domain of
From the above counter examples, we can claim that her approach is not valid.
Her approach is valid if and only if both and and are linear functions.
Alternative:
Let us start from an example.
Then,
Now if we examine domain of just by considering its formula for , certainly it will be whole number , also range will again be whole number . But as range for and excludes certain points in , so will . Hence it will not be valid to examine domain of a composite function just by taking its direct formula into account. In example above, domain for must be certainly .
7. Write yourself any two real-valued function. Find their composition.
Solution:
Let
Now,
Also,
8. A stone is thrown into a pond, creating a circular ripple that spreads over the pond in
such a way that the radius is increasing at the rate of 3 ft/sec.
(a) Find a function for the radius in terms of
Solution:
(b) Find a function for the area of the ripple in terms of the radius
Solution:
(c) Find Explain the meaning of this function.
Solution:
This function represents the area of circular ripple after second.