Question

Find the intercepts and asymptotes. (If an answer does not exist, enter DNE. Enter your asymptotes as a comma-separated list of equations if necessary.)

Also state the domain and range.

r(x) = (x − 1)(x + 3) / (x + 1)(x − 7)

Answer 1

La Given that, (x) (x-1)(x+3) (2+1) (x-7) # Now so (x) will be valid if (x+1)(x-7) & je. of x- and x 87 so, ventical asymprotes are and laat # NOW, lim xo n(x) = lim Gari) (243) (x+1)(x-7) = lim. (1+2) (17 %) x> (1+x) (1-7) (1-0)*(1+0) (1+0)*(1-0) 누 So, honizontal asymproti is asymprot is, ly = -)

# How x-intincept () is that point wherse y = n(x) 20 Now 2(x) = 0 implies -1) (x+3) = 0 = 2c - 2, so, x-intercepts asse ((-3,0) and ((1,0) # Now y-intincept is that point whesse a=0. *(0+3) X=0 implies, ro(o)- @+1) 0-7) so, yaintincept is (0, 2) # Domain is | (-0, -1) U (1,7) u (7,0) # Range is (0,0) Corde Now 3/4
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So we have,

x-intercepts : (-3, 0), (1, 0)

y-intercept : (0, 3/7)​​​​​​

Horizontal asymptote : y = 1​​​​​​

Vertical asymptotes : x = -1, x = 7​​​​​​

Domain : (-∞, -1) ∪ (-1, 7) ∪ (7, ∞)

Range : (-∞, ∞)