y=[(X^2+1)/(X^2-1)]+2?

Question

Any math wizard who can help me get the domain range asymtotes intercepts of this function. I will chose the best answer

Answer 1

asymptotes:
x = 1 & x = -1
x-axis intercepts x = +/- √(3)/3
y-axis intercept y = 1

Answer 2

y =

x² + 1
-------- + 2 =
x² - 1

x² - 1 + 2
------------- + 2 =
x² - 1

2
-------- + 3
x² - 1

#1.

y is undefined when x = -1 and x = 1.

Thus vertical asymptotes at x = -1 and x = 1

#2.

As x → ±∞, 2/(x² - 1) → 0 so y → 3

ie as x increases and decreases without bound y asymptotically approaches 3

#3.

When x = 0 y = -1 + 2 = 1 So the y-intercept is y = 1

#4.

When y = 0

2
-------- + 3 = 0
(x² - 1)

ie 2 + 3x² - 3 = 0
So 3x² = 1
ie x = ±√3/3
So it cuts the x-axis at x = -√3/3 and x = √3/3

#5.

dy/dx = -2x/(x² - 1)² = 0 when x = 0

So there is a stationary point at x = 0
When x < 0 dy/dx > 0 and when x > 0 dy/dx < 0

So there is a local maximum at (0, 1)

#6.

When x < -1, y >3
When x > 1, y > 3
When -1 < x < 1, y < 3

#7.

Domain: all x except x = ± 1
Range: y ≤ 1 and y > 3

That should be sufficient to get you started

Answer 3

x =-1 and 1

Answer 4

y= x^2+1 + 2(x^2) + 2 / (x^2-1)

y = [3 x^2 + 3 ]/ [(x-1)(x+1)]

asymptotes are : x = 1, x = -1