Work the Function f(x)=200x-15/40-2x...and g(x)=x^2/x-2...find asymptotes?

1)Work the Function f(x)=200x-15/40-2x

Find the Vertical, horizonal, or slant asymptotes?

and graph

2)Work the function g(x)=x^2/x-2

Find the Vertical, horizontal, or slant asymptotes?

and graph

1 Answer

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  • Wal C
    Lv 6
    1 decade ago
    Favorite Answer

    1. f(x) = 200x - 15/40 - 2x

    I assume you mean

    f(x) = (200x - 15)/(40 - 2x)

    = 5(40x - 3)/[2(20 - x)]

    = 5/2 * (40x - 3)/(20 - x)

    = 5/2 * (40x - 800 + 797)/(20 - x)

    = 5/2 * -40(20 - x)/(20 - x) +

    = 100 + 5/2 * 797/(20 - x)

    Now we can deal with it, as messy as it looks as before it was 'improper' since the degree of the numerator was the same as the degree of the denomonator and a rational function is 'improper' if degree numerator ≥ degree denominator

    Now lets look at 5/2 * 797/(20 - x)

    A. When x < 20, 20 - x > 0 so 5/2 * 797/(20 - x) > 0.

    Therefore f(x) = 100 + 5/2 * 797/(20 - x) (= (200x - 15)/(40 - 2x)) > 100

    AND when x > 20, 20 - x < 0 so 5/2 * 797/(20 - x) < 0.

    Therefore f(x) = 100 + 5/2 * 797/(20 - x) (= (200x - 15)/(40 - 2x)) < 100

    ie for x > 20, f(x) < 100 and for x < 20, f(x) > 100

    B. When x = 20 20 - x = 0 so 5/2 * 797/(20 - x) is undefined.

    Therefore f(x) = 100 + 5/2 * 797/(20 - x) (= (200x - 15)/(40 - 2x)) is undefined.

    As x → 20 from the left ,5/2 * 797/(20 - x) → +∞

    Therefore f(x) = 100 + 5/2 * 797/(20 - x) (= (200x - 15)/(40 - 2x))→ +∞

    AND as x → 20 from the right, 5/2 * 797/(20 - x) → -∞

    Therefore f(x) = 100 + 5/2 * 797/(20 - x) (= (200x - 15)/(40 - 2x))→ -∞

    ie vertical asympotes at x = 20

    C As x → ±∞, 5/2 * 797/(20 - x) → 0

    Therefore f(x) = 100 + 5/2 * 797/(20 - x) (= (200x - 15)/(40 - 2x)) → 100

    ie horitontal asymptotes at f(x) = 100

    D. You now have enough information to sketch this

    2. Work the function g(x)=x^2/x-2

    I assume you mean g(x)=x^2/(x - 2)

    Again this is improper as deg numerator = 2 and deg denominator = 1 (< deg numerator)

    So g(x) = x^2/(x - 2)

    = (x^2 - 4 + 4)/(x - 2)

    = ((x - 2)(x + 2) + 4)/(x - 2)

    = x + 2 + 4/(x - 2)

    Do steps A. to D as above

    A. Check out the signs of the denominator of the fraction

    When x < 2, x - 2 < 0 so 4/(x - 2) < 0

    Therefore g)x) = x + 2 + 4/(x - 2) {= x^2/(x - 2)} < x + 2

    When x > 2, x - 2 > 0 so 4/(x - 2) > 0

    Therefore g(x) = x + 2 + 4/(x - 2) {= x^2/(x - 2)} > x + 2

    ie for x < 2, g(x) < x + 2 and for x > 2, f(x) > x + 2

    B. When x = 2, x - 2 = 0 so 4/(x - 2) is undefined.

    Therefore g(x) = x^2/(x - 2) (= x + 2 + 4/(x - 2)) is undefined.

    As x → 2 from the left, 4/(x - 2) → -∞

    Therefore g(x) = x^2/(x - 2) (= x + 2 + 4/(x - 2)) → -∞

    AND as x → 2 from the right, 4/(x - 2) → +∞

    Therefore g(x) = x^2/(x - 2) (= x + 2 + 4/(x - 2)) → +∞

    ie vertical asymptotes at x = 2

    C. As x → ±∞, 4/(x - 2) → 0

    Therefore g(x) = x + 2 + 4/(x - 2) {= x^2/(x - 2)} → x + 2

    Therefore g(x) is assymptotic to g(x) = x + 2

    D. You now have enough information to sketch this

    Source(s): Me ;^))
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