Rational Function questions

Questions

1. Sketch 2x+1 / x-5 2. Sketch 4-x / 2x-3 3. Sketch 2x-1 / x+4 4. Solve the inequality x+5 / x+3 ≥ 2

Answer 1

X intercept: y=0 , 0=2x+1 , 2x=-1, x=-1/2 Y intercept: x=0 , y=1/-5 , y=-1/5 Vertical asymptope: x-5 = 0 , x=5 Horizontal asymptope: y=2+1/x / 1-5/x , as x heads to infinity, 1/x heads to 0, so the horizontal asymptope is 2/1, which is y=2 Slightly to left: f(4.99) = 2(4.99)+1 / 4.99 - 5 = -1098, which explains that to the left of the asymptope, the graph tends to the left Slightly to the right: f(5.01) = 2(5.01)+1 / 5.01 - 5 = 1102 which explains to the right of the asymptope, the graph tends to the right

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Answer 2

X intercept : x=4 Y intercept: y=-4/3 Vertical asymptope: x=3/2 Horizontal asymptope: y=-1/2 Slightly to left: f(1.49) = -125.5 (tends to left) Slightly to right: f(1.51) = 124.5 (tends to right)  

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Answer 3

X intercept : x=1/2 Y intercept: y=-1/4 Vertical asymptope: x=-4 Horizontal asymptope: y=2 Slightly to left: f(-4.01) = 902 (tends to right) Slightly to right: f(-3.99) = -898 (tends to left)

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Answer 4

We can multiply both sides by (x+3)^2 to get x^2+8x+15 > 2x^2+12x+18 , 0 > x^2+4x+3 , 0 > (x+1)(x+3) so x=-1 or -3, therefore -3<x≤-1.  Also we can use a graph to find this, we can see that y=2 intersects with the function at (-1,2) and the functions vertical asymptope is at x=-3, so -3<x≤-1. 

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