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Identity function is a function which gives the same value as inputted.

__
Example
__

f: X → Y

f(x) = x

Is an identity function

We discuss more about graph of f(x) = x in this post

##
Find identity function of
*
fog
*
and
*
gof
*

f: X → Y

& g: Y → X

*
gof
*

*
gof
*
= g(f(x))

gof : X → X

We input x

So, we should get x

∴
*
gof
*
= x

We write
*
gof
*
= I
_{
X
}

where I
_{
X
}
is the identity function in set X

**
fog
**

*
fog
*
= f(g(y))

fog : Y → Y

We input y

So, we should get y

∴ fog = y

We write fog = I
_{
Y
}

where I
_{
Y
}
is the identity function in set Y

##
**
Find Identity Function of set X and set Y
**

X = {1, 2, 3}

Y = {4, 5, 6}

In identity function,

we get what we input

If we input 1, we should get 1

####
**
Identity function of set X
**

If we input 1, we should get 1

If we input 2, we should get 2

If we input 3, we should get 3

∴ I
_{
X
}
= {(1, 1), (2, 2), (3, 3)}

**
Identity function of set
**
**
Y
**

If we input 4, we should get 4

If we input 5, we should get 5

If we input 6, we should get 6

∴ I
_{
Y
}
= {(4, 4), (5, 5), (6, 6)}