Answer 1:
[tex]\frac{x^{2} -x+1}{x^{2} +x+1}[/tex]
Here given that, x is real that is the domain
Range of the given expression is :
[tex]\frac{1}{3} \leq y\leq 3[/tex]
Write it interval form.
[tex][\frac{1}{3},3][/tex] Option (2)
Answer 3:
[tex]\frac{11x^{2} +12x+6}{x^{2} +4x+2}[/tex]
Here domain is all real x.
Range is :
{y ∈ R : y ≤ -5 or y ≥ 3}
In interval form:
(-∞, -5] ∪ [3, ∞)
So, y can't lies between -5 to 3 Option (2)
Answer 5 :
[tex]\frac{x^{2} -3x+4}{x^{2} +3x+4}[/tex]
Here domain is :
x is real
And range is :
{y ∈ R : [tex]\frac{-1}{7}[/tex] ≤ y ≤ 7}
We see that maximum y value is 7.
So, maximum value is 7. Option (4)
Answer 6:
[tex]\sqrt{9x^{2}+6x+1 } < (2 -x)[/tex]
Solve for x .
Square both the side
9x² + 6x + 1 = 4 + x² - 4x
8x² + 10x - 3 < 0
(2x + 3)(4x - 1) < 0
2x + 3 = 0 & 4x - 1 = 0
x = [tex]\frac{-3}{2}[/tex] & x = [tex]\frac{1}{4}[/tex]
So,
[tex]\frac{-3}{2} < x < \frac{1}{4}[/tex]
x ∈ [tex](-\frac{3}{2} , \frac{1}{4} )[/tex] Option (1)
