Question

range of the function f(x)=x^2+x+2/x^2+x+1

Answer 1

Answer:

y = x²−x+1/x²+x+1

Domain is x ∈R

For Range of function:

yx² + xy + y = x² - x + 1

(y-1)x² + (y+1)x + y-1 = 0

if y = -1

-2x² -1 -1 = 0

-2x² -2 = 0

Therefore, y cannot be equal to -1 since x will be a complex value

Now, y not equal to -1.

(y-1)x² + (y+1)x + y-1 = 0 has real roots

hence, (y+1)² - 4 (y-1)(y-1) ≥0

y² + 2y + 1 - 4y² + 8y - 4 ≥0

-3y² + 10y - 3 ≥ 0

3y² - 10y + 3 ≤0

3y² - 9y - y + 3 ≤ 0

3y (y-3) - 1 (y-3) ≤0

(3y-1)(y-3) ≤ 0

y ∈[1/3,3]

Hence, range of function is [1/3,3]

Answer 2

yx​2 + xy + y = x2 - x + 1

(y-1)x2 + (y+1)x + y-1 = 0

if y = -1

-2x2 -1 -1 = 0

-2x2 -2 = 0

Therefore, y cannot be equal to -1 since x will be a complex value

Now, y not equal to -1.  

(y-1)x2 + (y+1)x + y-1 = 0   has real roots

hence, (y+1)2 - 4 (y-1)(y-1) ≥0

y2 + 2y + 1 - 4y2 + 8y - 4 ≥0

-3y2 + 10y - 3 ≥ 0

3y2 - 10y + 3 ≤0

3y2 - 9y - y + 3 ≤ 0

3y (y-3) - 1 (y-3) ≤0

(3y-1)(y-3) ≤0

y ∈[1/3,3]

Hence, range of function is [1/3,3]