Math Top 3 PDF

v Topic 3 (3.6-3.7-3.8) – Polynomial, Power, and Rational Function Subtopic 3.6 Rational Functions ! 1. Describe how the graph of the function 𝑓(𝑥) = "#$ can be obtained by transforming the graph...

v Topic 3 (3.6-3.7-3.8) – Polynomial, Power, and Rational Function Subtopic 3.6 Rational Functions ! 1. Describe how the graph of the function 𝑓(𝑥) = "#$ can be obtained by transforming the graph % of the reciprocal function 𝑔(𝑥) = " A. Translate right 4 units, then stretch vertically by a factor of 3 B. Translate left 4 units, then stretch vertically by a factor of 3 C. Translate right 4 units, then translate down 3 units. D. Translate left 4 units, then translate up 3 units. 2. Refer to the graph given below to evaluate lim! 𝑓(𝑥). "→' A. 1 B. 2 C. ∞ D. −∞ "#( 3. Given 𝑓(𝑥) = " " )'")!* , which of the following is/are true? I. 𝑦 − intercept (0,0.2) II. vertical asymptotes: 𝑥 = −5 𝑎𝑛𝑑 𝑥 = 7 III. horizontal asymptotes: 𝑦 = 0 A. I and II only B. II and III only C. I and III only D. I , II and III Topic-3 Page 1 of 17 MAT50A  4. Which of the following represents the vertical asymptote for the function below? 10x − 2 f(x) = − 12x − 24 A. x = −2 B. x = 0 C. x=2 D. x = 10 +)$ 5. Use the function f(x) = )$+)%, to answer the below: i. State the points of discontinuity. ii. Domain: iii. Identify the horizontal and vertical asymptotes. '+" #%-+#%' 6. Use the function f(x) = +" #!+#' to answer the below: i. State the points of discontinuity. ii. Domain: iii. Identify the horizontal asymptote. iv. Which of the obtained points of discontinuity is a “hole” and which is a vertical asymptote? Topic-3 Page 2 of 17 MAT50A.(+) 7. f(x) = 1(+) , where p(x) is a polynomial of degree n and h(x) is a polynomial of degree k. Complete each statement in questions (A-C) with (=, , ≤, or ≥). A) f(x) has a horizontal asymptote if n ________ k. B) The x-axis is an asymptote of f(x) if n ________ k. C) f(x) doesn’t have a horizontal asymptote if n ________ k. +#! 8. Determine the asymptotes, if any, of f(x) = +" #%. A. Vertical: x = ±1 ; Horizontal: y = −3 B. Vertical: x = −3 ; Horizontal: y = ±1 C. No vertical; No horizontal D. No vertical; Horizontal: y = 0 %'+#* 9. Match the characteristics that describe the graph of f(x) = !+#2. horizontal asymptote ____ A. none vertical asymptote ____ B. x = –3 oblique asymptote ____ C. x = 3 D. y = 12 E. y = 4 F. y = –3 Topic-3 Page 3 of 17 MAT50A  10. Identify the features of the graph of f(x). Select all that apply. " # )%'* 𝑓(𝑥) = " " )'* A. There is a hole at x = 5. B. There is a vertical asymptote at x = –5 C. There is an oblique asymptote at y = x D. There is a horizontal asymptote at y = 1 E. There is a zero at x = 5 3(+)4)(+)5) 11. Identify the below in f(x) = (+)4)(+)6). vertical asymptote at x = ____ horizontal asymptote at y = ____ hole at x = ____17 12. Match each function with its graph. A B C +# )% f(x) = !+#! ____ +" )% g(x) = !+#! ____ +)% h(x) = !+#! ____ Topic-3 Page 4 of 17 MAT50A  13. Use function 𝐟(𝐱) to answer questions (A and B). 𝟐𝐱 𝟐 #𝐱 )𝟏 𝐟(𝐱) = 𝐱 𝟐 )𝟏 i. Find the asymptotes and intercepts. H.A.: V.A: y-intercept(s): x-intercept(s): '+" #+ )% ii. Use the intercepts and asymptotes found in part A to graph the function f(x) = +" )%. Show the asymptotes and intercepts on the graph. Topic-3 Page 5 of 17 MAT50A  !! "!#$ 14. Given the function f(x) =.Answer questions (i-ii). !! #% i. Find the intercepts. ii. Draw the graph of the function. (Hint: The asymptotes are already drawn; plot the intercepts and then sketch the graph) Topic-3 Page 6 of 17 MAT50A  𝟐𝐱 𝟐 )𝐱 )𝟏 15. Given the function 𝑓(𝑥) = 𝐱 𝟐 )𝟏. Answer questions (i-iv). i. Find the domain of f. ii. Identify the vertical and horizontal asymptotes. H.A.: V.A.: iii. At which point does the function has a hole? iv. Find the y-intercept. Topic-3 Page 7 of 17 MAT50A  +" )")' 16. Given the function 𝑓(x) = +)!. Answer (i-ii). i. Analyze the graph by finding the intercepts, asymptotes, domain, identify where the function is continuous, and use limits to describe the behavior at the asymptotes. 𝐲 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭: x-intercepts: Vertical asymptote: Explain why the function doesn't have a horizontal asymptote. Find the oblique asymptote. Domain: Continuity: End behavior: Topic-3 Page 8 of 17 MAT50A  ii. Sketch the graph. (Plot the intercepts, draw the asymptotes, then sketch the graph) Topic-3 Page 9 of 17 MAT50A  Subtopic 3.7 Rational Equations 1. Which of the following is/are solution(s) to the equation? ! 𝑥+" =4 A. 𝑥 = 1 𝑜𝑟 𝑥 = 3 B. Only 𝑥 = 3 C. Only 𝑥 = 1 D. Only 𝑥 = −1 ' % % 2. Solve: " " )% − ")% = ' A. 𝑥 = −3 B. 𝑥=1 C. 𝑥=0 D. No solution 3. Which of the following is/are solution(s) to the equation below? &' ) '"( + 𝑚 = '"( A. 𝑚 = −9 𝑜𝑟 𝑚 = 1 B. 𝑚 = −1 𝑜𝑟 𝑚 = 9 C. Only 𝑚 = −5 D. Only 𝑚 = 5 Topic-3 Page 10 of 17 MAT50A  4. Solve the below equation. Check for extraneous solutions. #* , $ + = +! "+ + +"% 5. Solve the below equation. Check for extraneous solutions. $ & $ +"( − +#& = +! "*+#%( Topic-3 Page 11 of 17 MAT50A  6. How much pure acid must be added to 50 mL of a 35% acid solution to produce a mixture that is 75% acid? Write the equation only. Don’t solve. Topic-3 Page 12 of 17 MAT50A  Subtopic 3.8 Solving Inequalities in One Variable % 1. Which of the following is the solution to (")$)" ≥0? A. All real numbers except 𝑥 = 4 B. All real numbers except 𝑥 = −4 C. All real numbers except 𝑥 = ±2 D. (4, +∞) 2. Solve the following polynomial inequality: (𝑥 − 4)(𝑥 + 2)(2𝑥 − 3) > 0 A. (−∞, −4) ∪ (−1.5, 2) B. (−2, 1.5) ∪ (4, +∞) C. (−2, 1.5) ∪ (−4, +∞) D. (−2, +∞) 3. Solve: (𝑥 ' + 1)(𝑥 ' − 2𝑥 + 8) ≥ 0 A. [−4,2] B. (−∞, −2] ∪ [4, ∞) C. (−∞, −4] ∪ [2, ∞) D. (−∞, ∞) " " )2 4. Solve: "#*