Given:f={(x, 1+x
2

x
2

​
):x∈R}
Domain=x
range=
1+x
2

x
2

​

We find different values of
1+x
2

x
2

​
for different values of x
Value of x value of
1+x
2

x
2

​

−2
1+(−2)
2

(−2)
2

​
=
1+4
4
​
=
5
4
​
=0.8
−1
1+(−1)
2

(−1)
2

​
=
1+1
1
​
=
2
1
​
=0.5
0
1+0
(0)
2

​
=0
1
1+(1)
2

(1)
2

​
=
1+1
1
​
=
2
1
​
=0.5
2
1+(2)
2

(2)
2

​
=
1+4
4
​
=
5
4
​
=0.8
Thus, value will always lie between 0 and 1
∴ the value of range of y=
1+x
2

x
2

​
is always positive
Also, it lies between 0 and 1
Hence, Range is any positive integer between 0 and 1 with 0 included.
Thus, Range of f=[0,1)