There are some special cases of a function which will be very useful. We discuss some of them below
(i) Constant function
(ii) Identity function
(iii) Real – valued function

**Special cases of function**

There are some special
cases of a function which will be very useful. We discuss some of them below

(i) Constant function

(ii) Identity function

(iii) Real – valued
function

A function *f *: *A
*→ *B *is called a constant function if the range of *f *contains only one element.
That is, *f *(*x *) = *c *, for all *x *∈ *A *and for some fixed *c *∈ *B.*

From Fig.1.37, *A *=
{*a*,*b*,*c*,*d*} and *B *= {1, 2, 3} and * f
*= {(*a*, 3),(*b*, 3),(*c*,
3),(*d*, 3)} . Since, *f *(*x*) = 3 for every *x *∈ *A *,
Range of *f *= {3} , *f
*is a constant function.

Let *A *be
a non–empty set. Then the function* f*: *A *→ *A* defined by *f *(*x*) = *x* for all *x *∈ *A *is called an identity function on *A *and is denoted by *I*_{A}.

If *A *= {*a*,*b*,*c*}
then *f*=*I _{A}* = {(

A function *f*: *A
*→ *B* is called a real valued function if the range of *f *is a subset of the
set of all real numbers **R** . That is, *f *(*A*) ⊆ **R**.

**Example 1.17**

Let *f *be a function from **R ** to **R** defined by *f *(*x*) = 3*x *− 5 . Find the
values of *a *and *b *given that (*a*, 4) and (1, *b*)
belong to *f*.

*Solution **f *(*x*) = 3*x *–
5 can be written as *f *= {(*x*,
3*x *– 5) | *x *∈ *R*}

(*a*, 4) means
the image of *a *is 4. That is, *f *(*a*) = 4

3a – 5 = 4 ⇒ *a *= 3

(1, *b*) means
the image of 1 is *b*.

That is, *f *(1)
= *b *⇒ *b *= −2

3(1) – 5 = *b *Þ *b *= –2

**Example 1.18 **The distance *S *(in
kms) travelled by a particle in time ‘*t*’ hours is given by *S*(*t*) = [* t*^{2} + *t *]*/2. *Find the distance
travelled by the particle after

(i) three and half
hours.

(ii) eight hours and
fifteen minutes.

** Solution **The distance travelled
by the particle in time

(i)* t *= 3.5 hours. Therefore

The distance travelled
in 3.5 hours is 7.875 kms.

*t *= 8.25 hours.
Therefore

The distance travelled
in 8.25 hours is 38.16 kms, approximately.

If the function *f *:
**R** → **R** is defined by ,

then find the values
of

(i) *f *(4)

(ii) *f *(-2)

(iii) *f *(4) + 2*f
*(1)

(iii) [*f *(1) +
3*f *(4)] / *f *(-3)

The function *f *is
defined by three values in intervals I, II, III as shown by the side

For a given value of *x
*= *a *, find out the interval at which the point *a *is located,
there after find *f *(*a*) using the particular value defined in that
interval.

(i) First, we see
that, *x *= 4 lie in the third interval.

Therefore, *f *(*x*)
= 3*x *− 2 ; *f *(4) = 3(4) – 2 = 10

(ii)* x *= −2 lies in the second interval.

Therefore, *f *(*x*)
= *x*^{2} – 2 ; *f *(−2) = (−2)^{2} – 2 = 2

(iii) From (i), *f *(4)
= 10 .

To find *f *(1),
first we see that *x *= 1 lies in the second interval.

Therefore, *f *(*x*)
= *x*^{2} – 2 ⇒ *f *(1) = 1^{2}
– 2 = −1

So, *f *(4) + 2*f
*(1) = 10 + 2(−1) = 8

(iv) We know that *f
*(1) = -1 and *f *(4) = 10.

For finding *f
*(-3), we see that *x *= −3 , lies in the first interval.

Therefore, *f *(*x*)
= 2*x *+ 7 ; thus, *f *(−3) = 2(−3) + 7 = 1

Hence

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