Answer
x-intercept: None
y-intercept: (0, 3/2)
Vertical Asymptote: x=-1
Domain: (-∞, -1) U (-1, ∞)
Horizontal Asymptote: y=1/2
Range: (1/2, ∞)
See graph below
Work Step by Step
$r(x)=\frac{x^2 + 2x +3}{2x^2 + 4x + 2}$
$x^2 + 2x +3= 0$
$x^2 + 2x + 1 = -3 + 1$
No x-intercept
y-intercept is the ratio of the constants, which is 3/2
Thus, the y-intercept is at (0, 3/2)
Vertical asymptotes are when the denominator is equal to 0
$2x^2 + 4x + 2 = 0$
$2 (x+1)^2 = 0$
x = -1
So, domain is from (-∞, -1) U (-1, ∞)
Horizontal asymptote is the ratio of the constants of the leading term (with equal degree)
1/2
Thus, the horizontal asymptote is at y=1/2
So the range is from (1/2, ∞)