Question

Let be a function from R into R . Determine the range of f .

Answer 1

The function f is defined as,

f={ ( x, x 2 1+ x 2 ):x∈R }

Let f( x )= x 2 1+ x 2

Put f( x )=y to find x in terms of y.

y= x 2 1+ x 2 y+y x 2 = x 2 y= x 2 −y x 2 y= x 2 ( 1−y )

Solve further.

x 2 = y 1−y x= y 1−y

As the root has to be positive or zero. So,

y 1−y ≥0

Therefore, y≥0 .

Denominator is also a root. But denominator cannot be 0 or negative. So,

1−y>0 y<1

From above conditions the range of function is positive integer, including 0, and approaches 1 .

Thus, the range of function is [ 0,1 )