Question

week 3 discussion

hi everyone and welcome to week 3 of math 102!

1. given the rational function

\\f(x) = \frac{-2x^3 + 22x^2 - 6x - 270}{3x^3 - 30x^2 + 21x + 54} = \frac{-2(x - 5)(x + 3)(x - 9)}{3(x + 1)(x - 2)(x - 9)}\\
a. find all holes. please give the x- and y-coordinates of the holes.
b. find all vertical asymptotes.
c. find the horizontal or oblique asymptote. (which is it?)
d. find all x-intercepts.
e. find the y-intercept. you do not have to graph the function.

1. given two rational functions, q and r, which of the following are rational functions?

a. \\(q + r\\)
b. \\(q - r\\)
c. \\(qr\\)
d. \\(\frac{q}{r}\\)
e. \\(q \circ r\\)

1. which was more difficult, exam 1 or exam 2? why?

1. what did you do better in preparing for exam 2 than for exam 1?

1. exponential growth and decay.

a. have you ever heard of exponential growth?
b. what about exponential decay?
c. what do those phrases mean to you?
d. can you think of an example of something that grows exponentially?

Explanation:

Step1: Identify common factors for holes

$$f(x) = \frac{-2(x - 5)(x + 3)(x - 9)}{3(x + 1)(x - 2)(x - 9)}$$

The common factor is \(x - 9\), so there is a hole at \(x = 9\).

Step2: Calculate the y-coordinate of the hole

$$\lim_{x \to 9} f(x) = \frac{-2(9 - 5)(9 + 3)}{3(9 + 1)(9 - 2)} = \frac{-2(4)(12)}{3(10)(7)} = -\frac{16}{33}$$

Step3: Find vertical asymptotes from simplified denominator

$$3(x + 1)(x - 2) = 0 \implies x = -1, x = 2$$

Step4: Find horizontal asymptote using degrees

Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients:

$$y = \frac{-2}{3}$$

Step5: Find x-intercepts from simplified numerator

$$-2(x - 5)(x + 3) = 0 \implies x = 5, x = -3$$

Step6: Find y-intercept by evaluating at zero

$$f(0) = \frac{-2(0 - 5)(0 + 3)}{3(0 + 1)(0 - 2)} = \frac{30}{-6} = -5$$

Step7: Determine closure properties of rational functions

Rational functions are closed under addition, subtraction, multiplication, division (except by the zero function), and composition. Thus, all options A, B, C, D, and E are rational functions.

Answer:

1. A. There is a hole at \((9, -\frac{16}{33})\).

B. The vertical asymptotes are \(x = -1\) and \(x = 2\).
C. The horizontal asymptote is \(y = -\frac{2}{3}\).
D. The x-intercepts are \((5, 0)\) and \((-3, 0)\).
E. The y-intercept is \((0, -5)\).

1. A. \(Q + R\), B. \(Q - R\), C. \(QR\), D. \(\frac{Q}{R}\), E. \(Q \circ R\) (All of them are rational functions).