Question

What is the range of the function x/x^2+1

Answer 1

Answer:

Maintain the following steps :-

1. Assume y = the given expression .

2. Cross multiplication, reduce the equation as a polynomial in terms of the variable used in the given expression (here x) .

3. The reduced equation will be a quadratic equation of x . Find the discriminant.

For a quadratic equation like ax² + bx + c = 0 , the discriminant is ( b² - 4ac ) .

4 . As x is real , the discriminant of the quadratic equation will be positive

i.e. Discriminant ≥ 0 .

5 . Discriminant contains y . Now solve the equation, 

Discriminant ≥ 0 to find the range of y .

Solution ::-

Let y = x / (x²+1) ,

=> x² y + y = x ,

=> x²y - x + y = 0 ,

Comparing this equation with ax² + bx + c = 0 , 

The discriminant is b²-4ac = (-1)² - 4 • y • y = 1 - 4y² .

[ where, a=y , b=-1 , c=y ]

Now, Discriminant = 1 - 4y² ≥ 0 ,

=> (1)² - (2y)² ≥ 0 ,

=> ( 1 + 2y )•( 1 - 2y ) ≥ 0 ,

=> ( 2y + 1 )•( 2y - 1 ) ≤ 0 ,

=> ( 2y + 1 ) ≥ 0 and ( 2y - 1 ) ≤ 0 ,

=> y ≥ -½ and y ≤ ½ ,

=> ½ ≥ y ≥ -½ .

So, the range of y is [ ½ , -½ ] .