Rational Functions PDF

GENERAL MATHEMATICS Rational Equations Graph of Rational Functions  GENERAL MATHEMATICS Define a rational function; Determine the domain and intercepts of rational functions; Sketch the graph of rational functions; Solve rational equations and inequalit...

GENERAL MATHEMATICS Rational Equations Graph of Rational Functions  GENERAL MATHEMATICS Define a rational function; Determine the domain and intercepts of rational functions; Sketch the graph of rational functions; Solve rational equations and inequalities 31-Aug-24 General Mathematics 2  GENERAL MATHEMATICS Determine a one-to-one function; Get the inverse of a given function; and Sketch the graph of the function and its inverse 31-Aug-24 General Mathematics 3  GENERAL MATHEMATICS RATIONAL FUNCTIONS A rational function is a ratio of two polynomial functions. That is, if 𝑝(𝑥) and 𝑞(𝑥) are polynomial functions and 𝑞(𝑥) ≠ 0, then 𝒑 𝒙 𝒇 𝒙 = 𝒒 𝒙 is a rational function. 31-Aug-24 General Mathematics 4  GENERAL MATHEMATICS Rational Equations Rational Inequalities Graph of Rational Functions  GENERAL MATHEMATICS RATIONAL EQUATION A rational equation is an equation involving rational functions. 31-Aug-24 General Mathematics 6  GENERAL MATHEMATICS Excluded values Values of x that make the rational expressions undefined. Extraneous solutions Solutions of rational equations without the excluding values 31-Aug-24 Lesson Title 7  GENERAL MATHEMATICS 1. Clear denominators by multiplying each term by the Least Common Denominator (LCD). 2. Simplify and solve familiar equation. 3. Check for excluded values and extraneous solutions. 31-Aug-24 General Mathematics 8  GENERAL MATHEMATICS −1 −2 6+𝑥 =𝑥 *The equation is from LibreTexts Mathematics (n.d.). 31-Aug-24 Lesson Title 9  GENERAL MATHEMATICS 6 + 𝑥 −1 = 𝑥 −2 1 1 6+ = 2 𝑥 𝑥 6𝑥 2 + 𝑥 − 1 = 0 3𝑥 − 1 2𝑥 + 1 = 0 3𝑥 − 1 = 0 2𝑥 + 1 = 0 𝟏 𝒙 = −𝟏/𝟐 𝒙= 𝟑 31-Aug-24 Lesson Title 10  GENERAL MATHEMATICS 5𝑥 + 22 𝑥+4 1+ 2 = 𝑥 + 3𝑥 − 4 𝑥 − 1 The equation is from LibreTexts Mathematics (n.d.). 31-Aug-24 Lesson Title 11  GENERAL MATHEMATICS 5𝑥 + 22 𝑥+4 1+ 2 = 𝑥−1 𝑥+4 𝑥 + 3𝑥 − 4 𝑥 − 1 𝑥 − 1 𝑥 + 4 + 5𝑥 + 22 = 𝑥 + 4 2 𝑥 2 + 3𝑥 − 4 + 5𝑥 + 22 = 𝑥 2 + 8𝑥 + 16 𝑥 2 + 8𝑥 + 18 = 𝑥 2 + 8𝑥 + 16 18 = 16 No solution 31-Aug-24 Lesson Title 12  GENERAL MATHEMATICS Domain – set of all real numbers Vertical Asymptotes – obtained from the value that makes the denominator zero. Horizontal asymptotes – x- 31-Aug-24 Graph | Asymptotes 13  GENERAL MATHEMATICS Vertical Horizontal Oblique / Slant Asymptotes in orange dash line (CueMath, n.d.) 31-Aug-24 Graph | Asymptotes 14  GENERAL MATHEMATICS Vertical Asymptote It is a line represented by the equation 𝑥 = 𝑎 where the numerator is zero and the denominator is non-zero. Removable discontinuity exists if the numerator and the denominator have the same factor. It appears as a hole in the graph. 31-Aug-24 Graph | Asymptotes 15  GENERAL MATHEMATICS Horizontal Asymptote Let 𝑓 be a rational function defined by the equation of two polynomials as follows: 𝑎𝑚 𝑥 𝑚 + ⋯ + 𝑎1 𝑥 + 𝑎0 𝑓 𝑥 = 𝑏𝑛 𝑥 𝑛 + ⋯ + 𝑏1 𝑥 + 𝑏0 𝒎𝒏 𝑎 𝑦=0 NO horizontal 𝑦= 𝑏 → Oblique 31-Aug-24 Graph | Asymptotes 16  GENERAL MATHEMATICS Oblique / Slant asymptote 𝑔 𝑥 If 𝑓(𝑥) = and the degree of 𝑔(𝑥) is more , ℎ(𝑥) than 1 the degree of ℎ(𝑥), then 𝑓(𝑥) can be expressed in the form 𝑟 𝑥 𝑓 𝑥 = 𝑚𝑥 + 𝑏 + ℎ(𝑥) The oblique asymptote is the line 𝑓(𝑥) = 𝑚𝑥 + 𝑏. 31-Aug-24 Graph | Asymptotes 17  GENERAL MATHEMATICS 1. Solve for the x-intercept, y-intercept, vertical asymptote, horizontal asymptote, oblique asymptote (conditional), and the domain. 2. Graph the asymptotes. 3. Plot the intercepts. 4. Create and complete the table of values. 31-Aug-24 Graph | Steps 18  GENERAL MATHEMATICS 2 𝑥 − 4𝑥 + 4 𝑓 𝑥 = 2 𝑥 −4 31-Aug-24 Lesson Title 19  2 𝑥 + 4𝑥 + 4 𝑓 𝑥 = 𝑥2 − 4 x-intercept: (−2,0) y-intercept: (0, −1) V.A.: 𝑥 = 1 H.A.: 𝑦 = 2 Domain: 𝑥 ∈ ℝ − {2} 𝒙𝟐 +𝟒𝒙+𝟒 Graph of 𝒇 𝒙 = 31-Aug-24 𝒙𝟐 −𝟒 Lesson Title 20