In this section, we will discuss the following types of functions with suitable examples.
(i) one – one
(ii) many – one
(iii) many – one
(iv) into

**Types of Functions**

In this section, we
will discuss the following types of functions with suitable examples.

(i) one – one

(ii) many – one

(iii) many – one

(iv) into

Let us assume that we
have a cell phone with proper working condition. If you make a usual call to
your friend then you can make only one call at a time (Fig.1.21).

If we treat making
calls as a function, then it will be one – one.

**A function f : A→ B is called **

A one-one
function is also called an injection.

Equivalently,

**If for all a_{1}, a_{2} **

**Illustration 10**

*A *= {1,2,3,4} and *B *=
{*a*,*b*,*c*,*d*,*e*}

(i) Let *f *=
{(1,*a*), (2,*b*), (3,*d*), (4,*c*)}

In Fig. 1.22, for
different elements in *A*, there are different images in *B*.

Hence *f *is a
one – one function.

(ii) Let *g *=
{(1,*a*), (2,*b*), (3,*c*), (4,*e*)}

* g *is a function from *A *to *B
*such that *g*(1) = *g*(2)
= *b *, but 1 ≠ 2.

Thus two distinct
elements 1 and 2 in the first set *A *have same image *b *the second
set in *B *(Fig.1.23). Hence, *g *is not a one–one function.

In a theatre complex
three films *F*_{1},
*F*_{2}* *, *F*_{3}* *are shown. Seven persons (*P*_{1} to *P* _{7})
arrive at the theatre
and buy tickets as shown (Fig 1.24).

If the selection of
films is considered as a relation, then this is a function which is many–one, since more
than one person may choose to watch the same film.

**A function f
: A**

In other words, a
function *f*: *A*→ *B* is called many-one if *f *it is not
one–one.

**Illustration 11**

Let *A *=
{1,2,3,4} and ; Let *f *= {(1,*a*), (2,*a*), (3,*b*), (4,*c*)}

Then *f *is a
function from *A *to *B *in which different elements 1 and 2 of *A *have
the same image *a *in *B*. Hence *f *is a many – one function.

In a mobile phone
assume that there are 3 persons in the contact. If every person in the contact
receives a call, then the function representing making calls will be onto.
(Fig.1.25)

**A function f
: A**

In other words, every
element in the co-domain *B *has a pre-image in the domain *A*. An
onto function is also called a **surjection.**

**note**

If f : A → B is an
onto function then, the range of *f* =
B . That is, *f* (A) = B .

**Illustration 12**

Let *A *= {*x*,*y*,
*z*}, *B *= {*l*,*m*,*n*} ;

Range of *f *= {*l*,*m*,*n*}
= *B *(Fig.1.26)

Hence *f *is an
onto function.

In a home appliance
showroom, the products television, air conditioner, washing machine and water
heater were provided with 20% discount as new year sale offer. If the selection
of the above products by the
three customers C_{1}, C_{2}, C_{3} is considered as a
function then the following diagram (Fig.1.27) will represent an into function.

During winter season
customers usually do not prefer buying air conditioner. Here air conditioner is
not chosen by any customer. This is an example of into function.

**A function f
: A**

That is the range of *f
*is a proper subset of the co-domain of *f*.

In other words, a
function *f*: *A*→ *B* is called ‘into’ if it is not ‘onto’.

**Illustration 13**

Let *A *= {1,2,3,}
and *B *= {*w*, *x*,*y*, *z*} ; *f *=
{(1,*w*),(2, *z*),(3,*x*)}

Here, range of *f
*= {*w*, *x*, *z*} ⊂ *B *(Fig.1.28)

Therefore, *f *is
a into function.

Note that *y *∈ *B* is not an image of any element in *A*.

Consider the circle
where each letter of the English alphabet is changed from
inner portion to a
letter in the
outer portion.

Thus *A *→ *D
*, *B *→ *E *, *C *→ *F *, … *Z *→ *C
*. We call this circle as‘cipher circle’. (Fig.1.29) In this way if we try
to change the word‘HELLO’ then it will become ‘KHOOR’. Now using the same
circle if we substitute for each outer
letter the corresponding inner letter we will get back the word ‘HELLO’. This
process of converting
from one form to an other form and receiving back
the required information is called **bijection**. This process is widely used in the study of secret
codes called ‘**cryptography**’.

*If a function f : A→ B is both one–one and onto,
then f is called a bijection from A to B.*

**Note**

A one – one and onto
function is also called a one – one correspondence.

To determine whether
the given function is one–one or not the following test may help us.

Previously we have
seen the vertical line test. Now let us see the horizontal line test. “A function represented in a graph is
one–one, if every horizontal line intersects the curve in at most one point”.

**Example 1.12 **

Using horizontal line
test (Fig.1.35(a), 1.35(b), 1.35(c)), determine which of the following
functions are one – one.

*Solution *

The curves in
Fig.1.35(a) and Fig.1.35(c) represent a one–one function as the horizontal
lines meet the curves in only one point *P*.

The curve in
Fig.1.35(b) does not represent a one–one function, since, the horizontal meet
the curve in two points *P *and *Q*.

**Example 1.13 **Let *A *=
{1, 2, 3} , *B *= {4, 5, 6, 7} and *f *= {(1,
4),(2, 5),(3, 6)}** **be a function from *A *to *B*. Show that *f *is
one – one but not onto function.

*Solution **A *= {1, 2, 3} , *B *=
{4, 5, 6, 7} ; *f *= {(1, 4),(2, 5),(3, 6)}

Then *f *is
a function from *A *to *B *and for different elements in *A*,
there are different images in *B*. Hence *f *is one–one function.
Note that the element 7 in the co-domain does not have any pre-image in the
domain. Hence *f *is not onto (Fig.1.36).

Therefore *f *is
one–one but not an onto function.

**Example 1.14 **If *A *=
{−2,−1, 0,1, 2} and* f*: *A*→ *B*
is an onto function defined by* f *(*x*) = *x*^{2} + *x
*+ 1 then find *B*.

** Solution **Given

*f *(-2) = (−2)^{2}
+ (−2) + 1 =3; *f *(-1) = (−1)^{2} + (−1) + 1 =1

*f *(0) = 0^{2} +
0 + 1 =1;* *

*f *(1) = 1^{2} + 1
+ 1 = 3

*f *(2) =2^{2} +
2+ 1 = 7

Since, *f *is an
onto function, range of *f *= *B *= co-domain of *f*.

Therefore, *B *=
{1, 3, 7}.

**Example 1.15** Let *f
*be a function *f *: **N** → **N** be defined by *f *(*x*) = 3*x *+ 2, *x *∈ N

(i) Find the images of
1, 2, 3

(ii) Find the pre-images
of 29, 53

(iii) Identify the
type of function

The function *f *:
**N** → **N** is defined by *f *(*x*)
= 3*x *+ 2

(i) If *x *=
1, *f *(1) = 3(1) + 2 = 5

If *x *= 2, *f *(2)
=3(2) + 2 = 8

If *x *= 3 , *f
*(3) = 3(3) + 2 = 11

The images of 1, 2, 3
are 5, 8, 11 respectively.

(ii) If *x *is
the pre-image of 29, then *f *(*x*) = 29 . Hence 3*x *+ 2 = 29

3*x *= 27 ⇒ *x *= 9.

Similarly, if *x *is
the pre-image of 53, then *f *(*x*) = 53 . Hence 3*x *+ 2 = 53 3*x
*= 51 ⇒ *x *= 17

Thus the pre-images of
29 and 53 are 9 and 17 respectively.

(iii) Since different
elements of **N** have different images in the co-domain, the function *f *is
one – one function.

The co-domain of *f *is
**N**.

But the range of *f *=
{5, 8, 11, 14, 17, ...} is a proper subset of **N**.

Therefore *f *is
not an onto function. That is, *f *is an into function. Thus *f *is
one – one and into function.

Forensic scientists
can determine the height (in cms) of a person based on the length of their
thigh bone. They usually do so using the function *h*(*b*) = 2 ⋅ 47*b *+ 54 ⋅ 10 where *b *is
the length of the thigh bone.

(i) Check if the
function *h *is one – one

(ii) Also find the
height of a person if the length of his thigh bone is 50 cms.

(iii) Find the length
of the thigh bone if the height of a person is 147 × 96 cms.

*Solution*

(i) To check if *h *is
one – one, we assume that *h*(*b*_{1}) = *h*(*b*_{2})

Then we get, 2 ⋅ 47*b*_{1} + 54 ⋅ 10 = 2 ⋅ 47*b*_{2} + 54 ⋅ 10

2 × 47*b*_{1}
= 2 ⋅ 47*b*_{2} ⇒ *b*_{1}
= *b*_{2}

Thus, *h*(*b*_{1})
= *h*(*b*_{2}) ⇒ *b*_{1}
= *b*_{2}. So, the function *h *is one – one.

(ii) If the length of
the thigh bone *b *= 50, then the height is* *

*h*(50) = (2 ⋅ 47 × 50) + 54 ⋅ 10 = 177 ⋅ 6 cms.

(iii) If the height of
a person is 147 × 96 cms, then *h*(*b*) = 147 ⋅ 96 and so the length of the thigh bone is given by 2 ⋅ 47*b *+ 54 ⋅ 10 = 147 ⋅ 96 .

*b *= [ 93 ⋅ 86 / 2 ⋅ 47 ] = 38

Therefore, the length of the thigh bone is 38 cms.

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