Multiple Choice Identify the choice that best completes the statement or answers the question. Find any points of discontinuity for the rational function. 1. \( y=\frac{(x-7)(x+2)(x-9)}{(x-5)(x-2)} \) a. \( x=-5, x=-2 \) b. \( x=5, x=2 \) c. \( x=-7, x=2, x=-9 \) d. \( x=7, x=-2, x=9 \) 2. \( y=\frac{(x+7)(x+4)(x+2)}{(x+5)(x-3)} \) a. \( x=-5, x=3 \) b. \( x=7, x=4, x=2 \) c. \( x=-7, x=-4, x=-2 \) d. \( x=5, x=-3 \) 3. \( y=\frac{x+4}{x^{2}+8 x+15} \) a. \( x=-5, x=-3 \) b. \( x=-4 \) c. \( x=-5, x=3 \) d. \( x=5, x=3 \) 4. \( y=\frac{x-3}{x^{2}+3 x-10} \) a. \( x=-5, x=2 \) b. \( x=5, x=-2 \) c. \( x=3 \) d. \( x \) \( =-5, x=-2 \) 6. What are the points of discontinuity? Are they all removable? \[ y=\frac{(x-4)}{x^{2}-13 x+36} \] a. \( x=-9, x=-4, x=8 \); yes b. \( x=1, x=8, x= \) -8; no c. \( x=9, x=4 \); no d. \( x=-9, x=-4 \); no 7. Describe the vertical asymptote(s) and hole(s) for the graph of \( y=\frac{(x-2)(x-5)}{(x-5)(x+2)} \). a. asymptote: \( x=2 \) and hole: \( x=-5 \) b. asymptotes: \( x=-2 \) and hole: \( x=-5 \) c. asymptote: \( x=-2 \) and hole: \( x=5 \) d. asymptote: \( x=-2 \) and hole: \( x=-2 \) a. \( x=-3, x=-8 \); no b. \( x=5, x=-7, x=1 \); no c. \( x=-5, x=7, x=-1 \); yes d. \( x=3, x=8 \); yes 8. Describe the vertical asymptote(s) and hole(s) for the graph of \( y=\frac{x+1}{x^{2}+6 x+5} \). a. asymptote: \( x=-1 \) and hole: \( x=-1 \) b. asymptote: \( x=-1 \) and hole: \( x=-1 \) c. asymptotes: \( x=-1,-1 \) and hole: \( x=-1 \) d. asymptote: -5 and hole: \( x=-1 \), 9. Find the horizontal asymptote of the graph of \( y=\frac{7 x^{6}+7 x+3}{9 x^{5}+7 x+3} \). a. \( y=0 \) b. \( y=\frac{7}{9} \) c. no horizontal asymptote d. \( y=\frac{6}{5} \)

Sure, let's go through each question one by one to identify the correct answers. --- ### **1.** \( y = \frac{(x-7)(x+2)(x-9)}{(x-5)(x-2)} \) **Points of Discontinuity:** - **Denominator Zeros:** \( x = 5 \) and \( x = 2 \) - **Numerator Factors:** \( x = 7, -2, 9 \) (No cancellation with denominator) **Answer:** **b. \( x = 5, x = 2 \)** --- ### **2.** \( y = \frac{(x+7)(x+4)(x+2)}{(x+5)(x-3)} \) **Points of Discontinuity:** - **Denominator Zeros:** \( x = -5 \) and \( x = 3 \) - **Numerator Factors:** \( x = -7, -4, -2 \) (No cancellation with denominator) **Answer:** **a. \( x = -5, x = 3 \)** --- ### **3.** \( y = \frac{x+4}{x^{2} + 8x + 15} \) **Points of Discontinuity:** - **Simplify Denominator:** \( x^{2} + 8x + 15 = (x+5)(x+3) \) - **Denominator Zeros:** \( x = -5 \) and \( x = -3 \) - **Numerator Factor:** \( x = -4 \) (No cancellation with denominator) **Answer:** **a. \( x = -5, x = -3 \)** --- ### **4.** \( y = \frac{x-3}{x^{2} + 3x - 10} \) **Points of Discontinuity:** - **Simplify Denominator:** \( x^{2} + 3x - 10 = (x+5)(x-2) \) - **Denominator Zeros:** \( x = -5 \) and \( x = 2 \) - **Numerator Factor:** \( x = 3 \) (No cancellation with denominator) **Answer:** **a. \( x = -5, x = 2 \)** --- ### **6.** \( y = \frac{(x-4)}{x^{2} - 13x + 36} \) **Points of Discontinuity:** - **Simplify Denominator:** \( x^{2} - 13x + 36 = (x-9)(x-4) \) - **Denominator Zeros:** \( x = 9 \) and \( x = 4 \) - **Numerator Factor:** \( x = 4 \) (Cancels with denominator, creating a hole) - **Non-Removable Discontinuity:** \( x = 9 \) **Are All Discontinuities Removable?** No, \( x = 9 \) is not removable. **Answer:** **c. \( x = 9, x = 4 \); no** --- ### **7.** \( y = \frac{(x-2)(x-5)}{(x-5)(x+2)} \) **Vertical Asymptotes and Holes:** - **Simplify:** \( y = \frac{(x-2)\cancel{(x-5)}}{\cancel{(x-5)}(x+2)} = \frac{x-2}{x+2} \) - **Hole:** At \( x = 5 \) (where \( x-5 \) was canceled) - **Vertical Asymptote:** At \( x = -2 \) (denominator zero) **Answer:** **c. asymptote: \( x = -2 \) and hole: \( x = 5 \)** --- ### **8.** \( y = \frac{x+1}{x^{2} + 6x + 5} \) **Vertical Asymptotes and Holes:** - **Simplify Denominator:** \( x^{2} + 6x + 5 = (x+1)(x+5) \) - **Simplify Expression:** \( y = \frac{\cancel{(x+1)}}{\cancel{(x+1)}(x+5)} = \frac{1}{x+5} \) - **Hole:** At \( x = -1 \) - **Vertical Asymptote:** At \( x = -5 \) **Answer:** **d. asymptote: \( x = -5 \) and hole: \( x = -1 \)** --- ### **9.** \( y = \frac{7x^{6} + 7x + 3}{9x^{5} + 7x + 3} \) **Horizontal Asymptote:** - **Degree of Numerator:** 6 - **Degree of Denominator:** 5 - **Rule:** If the degree of the numerator is greater than the degree of the denominator, **no horizontal asymptote** exists. **Answer:** **c. no horizontal asymptote** --- ## **Summary of Answers:** 1. **b.** \( x = 5, x = 2 \) 2. **a.** \( x = -5, x = 3 \) 3. **a.** \( x = -5, x = -3 \) 4. **a.** \( x = -5, x = 2 \) 6. **c.** \( x = 9, x = 4 \); no 7. **c.** asymptote: \( x = -2 \) and hole: \( x = 5 \) 8. **d.** asymptote: \( x = -5 \) and hole: \( x = -1 \) 9. **c.** no horizontal asymptote --- If you have any further questions or need explanations for any of the problems, feel free to ask!