Let f={(x,x21+x2):x∈R} be a function from R into R. Determine the range of f.
Let y=x21+x2
⇒y+x2y=x2
⇒y=x2(1−y)
⇒x2=y1−y
⇒x=√y1−y
Since, x is real
⇒y1−y≥0
⇒y(1−y)(1−y)2≥0
⇒y(1−y)≥0 and (1−y)2>0
⇒0≤y≤1 and −y>−1
⇒0≤y≤1 and y<1
Hence, 0≤y<1
Range of f is [0,1)