For each of the following rational functions: Find the domain. Find the vertical

Question

For each of the following rational functions: Find the domain. Find the vertical asymptote(s), if any. If no vertical asymptote, write "No VA". State whether the function has a horizontal asymptote, a slant asymptote or neither; then state the equation of the horizontal or slant asymptote, if any. A(x) = 5x + 1/2 + 7x B(x) = x + 8/x^2 - 9 C(x) = 2x^3 - 7x^2 - 8x + 11/x^2 - 6x D(x) = x^4/(4 - 5x)(1 - x) E(x) = 5x^2/x^2 + 1 A rare species of insect was discovered in the rain forest. In order to protect the species, environmentalists declare the insect endangered and transplant the insects into a protected area. The population of the insect t months after being transplanted is given by P(t). P(t) = 60(1 + 0.4 t)/(0.01t + 3) How many insects were discovered? In other words, what was the population when t = 0? What will the population be after 5 years? Determine the horizontal asymptote of P (t). What is the largest population that the protected area can sustain? Graph P (t).

Explanation / Answer

1) A(x) = (5x +1)/(2 +7x)

Domain : 2+7x =0 ; x= -2/7

So, ( - inf , -2/7) U ( -2/7 , inf)

Vertical asmyote : x = -2/7

Horizontal asymtote : y = 5x/7x = 5/7

y = 5/7

B(x) = (x+8)/(x^2 -9) = (x+8)/(x+3)(x -3)

Domain : (x+3)(x -3) =0

( -inf, -3) U ( -3, 3) U( 3, inf)

Vertical asymtote : x= -3 ; x=3

Horizontal asymtote: if degree of denominator is less than the degrees of numerator, then

x axis is the horizontal asymtote.

y =0

C(x) = (2x^3 - 7x^2 - 8x +11)/(x(x-6)

Domain : x(x-6) =0

x =0 ; x=6

Vertical asymtote : x=0 ; x=6

Horizontal asymtote :If degree of numerator is greater than degree of denominator then no horizontal

asymtote exists ,but slant asymtote exist.

Slant asymtote : y = divide numerator by denomintor = (2x^3 - 7x^2 - 8x +11)/(x(x-6) = 2x +5

D(x) = x^4/( 4 - 5x)(1 -x)

Domain: 4 -5x =0; 1-x =0

x = 4/5 ; x= 1

( - inf , 4/5) U (4/5 , 1) U ( 1, inf)

Vertical asymote :x = 4/5 ; x=1

Horizontal asymtot ::If degree of numerator is greater than degree of denominator then no horizontal

asymtote exists .

If numerator degree > 1+ degree of denominator . no horizontal asymtote exist

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