Question

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?
e. can you think of an example of something that decays exponentially?

Explanation:

Step1: Identify common factors for holes

$$x - 9 = 0 \implies x = 9$$

Step2: Calculate y-coordinate of the hole

$$y = \frac{-2(9 - 5)(9 + 3)}{3(9 + 1)(9 - 2)} = -\frac{16}{15}$$

Step3: Find vertical asymptotes from remaining denominator

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

Step4: Determine horizontal asymptote using leading coefficients

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

Step5: Find x-intercepts from remaining numerator

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

Step6: Calculate y-intercept by setting x to 0

$$f(0) = \frac{-270}{54} = -5$$

Step7: Determine closure properties of rational functions

All operations A, B, C, D, and E yield rational functions.

Answer:

1.
A. Hole at \((9, -\frac{16}{15})\)
B. Vertical asymptotes at \(x = -1\) and \(x = 2\)
C. Horizontal asymptote at \(y = -\frac{2}{3}\)
D. x-intercepts at \((5, 0)\) and \((-3, 0)\)
E. y-intercept at \((0, -5)\)

1. A. \(Q + R\), B. \(Q - R\), C. \(QR\), D. \(\frac{Q}{R}\), E. \(Q \circ R\)