University:
University of Michigan
Course:
MATH 115 | Calculus I
Academic year:

2024
Views:

102

Pages:

2
Author:

Luciano Kirby

Question 1 True or False. Some rational functions have holes in their graph. Answer Some rational functions have holes in their graph. This is true, the values that makes denominator is hole. Question 2 How do exponential functions differ from polynomial and rational functions? Answer For example, it differ from polynomial in next polynomial functions have no asymptotes but exponential functions have, it is x-axis. When we are talking about difference between rational and exponential functions, one difference is in domain. Domain of exponential functions is set IR, but domain of rational function exclude those points which are zeros of denominator. And also, because of that, in those points, rational function has vertical asymptotes, but exponential function has no vertical asymptotes. Question 3 Write equations of two rational functions with graphs that are identical except that one is shifted vertically 3 units with respect to the other. Answer "Possible answers: f(x) = 1/x & g(x) = 1/x +3 f(x) = 1/(x+2) & g(x) = 1/(x+2) +3 f(x) = (x-1)/(x-1) & g(x) = (x-1)/(x-1) +3 Question 4 For the following exercises, find the domain of the rational functions. f(x) = (x-1)/(x+2) Answer Function: f(x) = (x-1)/(x+2) For a function that is defined by a fraction, the function is undefined when its denominator is equal to 0, therefore equate the denominator with 0 to evaluate the value of x for which the function will be undefined: x = -2 Therefore, it can be concluded that the domain of the function are all real numbers except -2. Question 5 Describe the key characteristics of the graphs of rational functions of the form f(x) = (ax + b) / (cx + d). Explain how you can determine these characteristics using the equations of the functions. In what ways are the graphs of all the functions in this family alike? In what ways are they different? Use examples in your comparison. Answer This kind of function will have vertical asymptote at x = -(d/c), horizontal asymptote at y = a/c. Its points of untersects will be: (0,b/d) with y-axis and (-(b/a), 0) with x-axis For example, let's take following function for values a=2, b=4, c=1. d=3: y = (2x+4)/(x+3) This function will have: vertical asymptote" x = -(d/c) = -(3/1) = -3 horizontal asymptote: y = a/c = 2/1 = 2 y-intercept: y = b/d = 4/3 x-intercept: x = -(b/a) = -(4/2) = -2