PROBLEMS AND SOLUTIONS - RATIONAL FUNCTIONS
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1 PROBLEMS AND SOLUTIONS - RATIONAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE THAT YOU CANNOT ALWAYS USE A CALCULATOR ON THE ACCUPLACER - COLLEGE-LEVEL MATHEMATICS TEST! YOU MUST BE ABLE TO DO SOME PROBLEMS WITHOUT A CALCULATOR! Problem 1: find the following:, which is also called the Reciprocal Function, Problem 2: Problem 3: a. the Domain in Interval Notation
2 Problem 4: Problem 5: a. the Domain in Interval Notation Problem 6: Problem 7: a. the Domain in Interval Notation
3 Problem 8: Problem 9: Problem 10: Problem 11:
4 Problem 12: Problem 1: SOLUTIONS You can find detailed solutions below the link for this problem set! Equation of the Vertical Asymptote: x = 0 Equation of the Horizontal Asymptote: y = 0 Please note that the branches of rational functions have SMOOTH turns. They are never parallel to their asymptotes, but move toward them at a steady pace. Please note that the x-axis is a horizontal asymptote and the y- axis is the vertical asymptote. Problem 2: Equation of the Vertical Asymptote: None Equation of the Horizontal Asymptote: y = 0
5 Please note that the x-axis is a horizontal asymptote. There are no vertical asymptotes! Remember that graphs may cross/touch their horizontal asymptotes! Problem 3: Equation of the Vertical Asymptote: and Equation of the Horizontal Asymptote: y = 4 There is a horizontal asymptote at y = 4 and two vertical asymptotes at and. Please note that asymptotes are invisible lines and NO graphing program places lines into a graph to represent asymptotes. Problem 4: Equation of the Vertical Asymptote: None Equation of the Horizontal Asymptote: None
6 Please note that there are no asymptotes! Problem 5: Equation of the Vertical Asymptote: and Equation of the Horizontal Asymptote: y = 2 There is a horizontal asymptote at two vertical asymptotes at and. Remember that graphs may cross/touch their horizontal asymptotes! and Problem 6: Equation of the Vertical Asymptote: None Equation of the Horizontal Asymptote: y = 2
7 There is a horizontal asymptote at no vertical asymptotes. and Problem 7: Equation of the Vertical Asymptote: Equation of the Horizontal Asymptote: y = -1 There is a horizontal asymptote at and a vertical asymptote at. Problem 8: Equation of the Vertical Asymptote: x = 2 Equation of the Horizontal Asymptote: None Equation of the Oblique Asymptote:
8 There is a vertical asymptote at and an oblique asymptote at. Problem 9: Equation of the Vertical Asymptote: x = 2 Equation of the Horizontal Asymptote: y = 0 Coordinates of any Holes: Please note that the x-axis is a horizontal asymptote. There is also a vertical asymptote at. Please observe the hole in the graph at. Problem 10: Equation of the Vertical Asymptote: x = 5 Equation of the Horizontal Asymptote: Coordinates of any Holes:
9 There is a horizontal asymptote at and a vertical asymptote at. Please observe the hole in the graph at. Problem 11: Equation of the Vertical Asymptote: x = -1 Equation of the Horizontal Asymptote: y = 0 Coordinates of any Holes: There is a horizontal asymptote at y = 0 and a vertical asymptote at x = -1. Please observe the hole in the graph at. Problem 12: Equation of the Vertical Asymptote: None Equation of the Horizontal Asymptote: None Coordinates of any Holes: (1, 2)
10 There is NO horizontal or vertical asymptote. Please observe the hole in the graph at (1, 2).
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