Section 2.3 (e-book 4.1 & 4.2) Rational Functions Definition 1: The ratio of two polynomials is called a rational function, i.e., a rational function has the form, where both and are polynomials. Remark 1: In this course we consider only the rational functions where and are relatively prime. Example 1: The following are rational functions and many more... Remark 2: a) All real numbers and also all polynomials are rational functions (Why?) b) There are functions which are not rational functions. We will study two of such functions, i.e, the exponential and logarithmic functions, in the next chapter. Example 2: For the rational functions and, evaluate a) b) c) d)
63 Domain of a Rational Function: Let be a rational function. Then D( f ) = { x : is defined}= {x : } = all real numbers except the zeros of. Example 3: Find the domain of each of the following rational functions a) b) b) y-intercept of Rational Functions: As before, the y-intercept is obtained by setting y-intercept =, i.e.,. Example 4: Find the y-intercept of the given functions a) b)
x-intercept of a Rational Function: The x-intercepts are obtained by setting and solving for x. This is equivalent to setting the numerator (why?) Example 5: Find the x-intercepts of the following rational functions 64 a) b) c) Definition 2: An asymptote of a function is a line that the graph of the function approaches. This is equivalent to saying that the distance between the graph of the function and its asymptote gets smaller and smaller, i.e., the distance approaches zero. An asymptote may be horizontal, vertical, slant, or any other type of curve. The asymptotes of a function give information about the end behavior of the function.
Remark 4: All rational functions except the polynomials, have asymptotes. The type of asymptote of a rational function depends on the degree of and. Remark 5: In this course, we will study the rational functions with vertical and\or the horizontal asymptotes only. Definition 3: a) The line is a vertical asymptote of a function if as x approaches a from left or right, the value approaches or. b) The line is a horizontal asymptote of a function if as x approaches or, the value approaches. 65 Finding Vertical Asymptotes: Theorem 1: Let be a rational function. A vertical line is a vertical asymptote if and only if. Another words, a vertical asymptote occurs at a real zero of the denominator. Remark 6: The domain of a rational function consists of the segments of the x-axis separated by the vertical asymptotes.
Example 6: Find and graph the vertical asymptotes of the functions below. Also mark the domain of the function on the x-axis 66 a) b) c) d)
67 Finding Horizontal Asymptotes Theorem 2: A rational function has at most one horizontal asymptote. a) If the degree of < degree of, then the line (x-axis) is the horizontal asymptote. b) If the degree of = degree of, then the line the ratio of the leading coefficients of and is the horizontal asymptote. c) If the degree of > degree of, then the rational function has no horizontal asymptotes. In this case the function has slant, parabolic, or other forms of asymptotes. We will not study these cases in this course. Examples 7: Find and sketch the horizontal asymptotes of the given functions and. a) c) d)
68 Guideline for Sketching Graphs of Rational Functions: Step 1: Find the domain, the vertical asymptotes as well as the horizontal asymptote. Step 2: Find the y-intercept. Step 3: Find the x-intercepts. Step 4 : Construct a table of (a few extra) values if necessary. Step 5: Transfer all the information above to an xy-plane and draw the graph. Step 6: verify your result by sketching the graph in a calculator. Example 8: Sketch the graph of the functions given below b) c)
69 d) Application: For a person with sensitive skin, the amount of time (in hours) the person can be exposed to the sun with minimal burning can be modeled by where s is the sunsor Scale reading. The Sunsor Scale reading is based on the level of intensity of UVB ray. i) find the amount of time a person with sensitive skin can be exposed to the sun with minimal burning when and., ii) What would the horizontal asymptote of this function represent?
70 Exercise: 1. Consider the rational function. a) Find the domain of the function. b) Find the vertical asymptotes of the function. c) Find the horizontal asymptote. d) Find the y-intercept of the function. e) Find the zeros of the function. f) Sketch the graph of the function. 2. Sketch the graph of the function given below 3. The population of a community is given by the formula, where t represents the time in decades. a) Find the population in 2, 5, 10, 15, 20 decades b) What is the long run population of this community? Explain