MAC 1140 Module 6 Nonlinear Functions and Equations II Learning Objectives Upon completing this module, you should be able to 1. identify a rational function and state its domain.. find and interpret vertical asymptotes. 3. find and interpret horizontal asymptotes. 4. solve rational equations. 5. solve applications involving rational equations. 6. solve applications involving variations. 7. solve polynomial inequalities. 8. solve rational inequalities. Rev.S10 Learning Objectives (Cont.) 9. learn properties of rational exponents. 10. learn radical notation. 11. use power functions to model data. 1. solve equations involving rational exponents. 13. solve equations involving radical expressions. Rev.S10 3 http://faculty.valenciacc.edu/ashaw 1
Nonlinear Functions and Equations II There are three sections in this module: 4.6 Rational Functions and Models 4.7 More Equations and Inequalities 4.8 Radical Equations and Power Functions Rev.S10 4 What is a Rational Function? A rational function is a nonlinear function. The domain of a rational function includes all real numbers except the zeros of the denominator q(x). Rev.S10 5 What is a Vertical Asymptote? In this graph, the line x is a vertical asymptote. Rev.S10 6 http://faculty.valenciacc.edu/ashaw
What is a Horizontal Asymptote? In this graph, the line y 5 is a horizontal asymptote. Rev.S10 7 Use the graph of Let s Look at an Example 1 to sketch the graph of x 1 g( x). ( x + 1)! Include all asymptotes in your graph. Write g(x) in terms of f(x). g(x) is a translation of f(x) left one unit and down units. The vertical asymptote is x 1 The horizontal asymptote is y g(x) f(x + 1) Rev.S10 8 How to Find Vertical and Horizontal Asymptotes? Rev.S10 9 http://faculty.valenciacc.edu/ashaw 3
More Examples For each rational function, determine any horizontal or vertical asymptotes. a) x + 6 b) x! 1 c) x! 4 4x! 8 x! 9 x! To identify horizontal asymptote, look at the leading coefficient of the highest power term in both numerator and denominator. a) Horizontal Asymptote: If the Degree of numerator equals the degree of the denominator, y a/b is asymptote, so y /4 1/ To identify vertical asymptote, set the denominator to 0. Vertical Asymptote: 4x 8 0, x Rev.S10 10 Examples (Cont.) For each rational function, determine any horizontal or vertical asymptotes. x + 6 x! 1 x! 4 a) b) c) 4x! 8 x! 9 x! (Cont.) b) Horizontal Asymptote: Degree: numerator < denominator y 0 is the horizontal asymptote. Vertical Asymptote: x 9 0 x ± 3 are the vertical asymptotes. Rev.S10 11 (Cont.) Examples (Cont.) For each rational function, determine any horizontal or vertical asymptotes. a) x + 6 b) x! 1 c) x! 4 4x! 8 x! 9 x! c) Horizontal Asymptote: Degree: numerator > denominator no horizontal asymptotes x! 4 Vertical Asymptote: x! no vertical asymptotes ( x! )( x + ) (There will be a hole x! in the graph.) x + x " The graph is the line y x + with the point (,( 4) missing. Rev.S10 1 http://faculty.valenciacc.edu/ashaw 4
What is a Slant/Oblique Asymptote? A third type of asymptote is neither horizontal or vertical. Occurs when the numerator of a rational function has a degree one more than the degree of the denominator. Rev.S10 13 Let x + 1. x! a) Use a calculator to graph f. Example b) Identify any asymptotes. c) Sketch a graph of f that includes the asymptotes. a) Reminder: Slant/Oblique Asymptote occurs when the numerator of a rational function has a degree one more than the degree of the denominator. Is that true in this rational function? Rev.S10 14 (Cont.) Example (Cont.) b) Asymptotes: The function is undefined when x 0 or when x. * Vertical asymptote at x * Oblique asymptote at y x + c) How about horizontal asymptote? Why don t we have it in this rational function? How can you tell from the function itself? Rev.S10 15 http://faculty.valenciacc.edu/ashaw 5
Solve How to Solve Rational Equation? x 4. x! Symbolic Graphical Numerical x 4 x! x 4( x! ) x 4x! 8! x! 8 x 4 Rev.S10 16 One More Example Solve 3 4 +. x! 1 x! 1 x + 1 Multiply by the LCD to clear the fractions. 4 3 + x! 1 x! 1 x + 1 4( x! 1)( x + 1) ( x! 1)( x + 1) 3( x! 1)( x + 1) + x! 1 x! 1 x + 1 4 ( x + 1) + 3( x! 1) 4 x + + 3x! 3 4 5x! 1 1 x When 1 is substituted for x,, two expressions in the given equation are undefined. There are no solutions. Rev.S10 17 Direct Variation The nonzero number k is called the constant of variation or the constant of proportionality. In the area formula for a circle, A π r, the π is the constant of variation; so, k π in this case. Rev.S10 18 http://faculty.valenciacc.edu/ashaw 6
Inverse Variation This inverse variation occurs when we have two quantities that vary inversely; the increase of one quantity will decrease the other quantity. Rev.S10 19 Example of Application At a distance of 3 meters, a 100-watt bulb produces an intensity of 0.88 watt per square meter. a) Find the constant of proportionality k. b) Determine the intensity at a distance of.5 meters. a) Substitute d 3 and I 0.88 into the equation and solve for k. 7.9 b) Let I and d.5. d 7.9 I d 7.9 I 1.7.5 k I d k 0.88 or k 7.9 3 The intensity at.5 meters is 1.7 watts per square meter. Rev.S10 0 How to Solve Polynomial Inequalities? An inequality says that one expression is greater than, greater than or equal to, less than, or less than or equal to, another expression. To solve Polynomial Inequalities, we need the following: Boundary numbers (x-values) are found where the inequality holds. A graph or a table of test values can be used to determine the intervals where the inequality holds. Rev.S10 1 http://faculty.valenciacc.edu/ashaw 7
Solving Polynomial Inequalities in Four Steps Rev.S10 Let s Look at This Example 3 Solve x! " 7x " 10x symbolically and graphically. Symbolically 3 Step 1: Write the inequality as x + 7x + 10x! 0. Step : Replace the inequality symbol with an equal sign and solve. x x x 3 + 7 + 10 0 ( x ) ( )( x ) x x + 7 + 10 0 x x + 5 + 0 x 0 or x! 5 or x! The boundary numbers are 5,, and 0. Rev.S10 3 Let s Look at This Example (Cont.) Step 3: The boundary numbers separate the number line into four disjoint intervals:!",! 5,! 5,!,!,0, and 0, " ( ) ( ) ( ) ( ) -8 - -6-4 0 4-7 -5-3 -1 1 3 5 Rev.S10 4 http://faculty.valenciacc.edu/ashaw 8
Let s Look at This Example (Cont.) Step 4: Complete a table of test values. Interval Test Value x x 3 + 7x + 10x Positive/Negative 6 4 4 8 1 4 1 18 [ ) The solution set is [! 5,! ]! 0, ". Negative Positive Negative Positive Rev.S10 5 Let s Look at This Example (Cont.) Graphically: Rev.S10 6 How to Solve Rational Inequalities? Inequalities involving rational expressions are called rational inequalities. Rev.S10 7 http://faculty.valenciacc.edu/ashaw 9
Example 5 Solve! 1. x + 4 Step 1: Rewrite the inequality in the form 5! 1 x + 4 5 " 1! 0 x + 4 5 "( x + 4)! 0 x + 4 1" x! 0 x + 4 p( x) 0. q( x )! Rev.S10 8 Example (Cont.) Step : Find the zeros of the numerator and the denominator. Numerator 1 x 0 x 1 Denominator x + 4 0 x 4 Step 3: The boundary numbers are 4 and 1, which separate the number line into three disjoint intervals: (!",! 4 ),(! 4,1 ) and ( 1, "). Rev.S10 9 Example (Cont.) Step 4: Use a table to solve the inequality. Interval Test Value x (1 x)/(x + 4) Positive/Negative 5 6 Negative 3/ Positive 1/6 Negative The interval notation is ( 4, 1]. Caution: When solving a rational inequality, it is essential not to multiply or divide each side of the inequality by the LCD if the LCD contains a variable. This techniques often leads to an incorrect solution set. Rev.S10 30 http://faculty.valenciacc.edu/ashaw 10
Let s Review Some Properties of Rational Exponents Rev.S10 31 Let s Practice Some Simplification Simplify each expression by hand. a) 8 /3 b) ( 3) 4/5 s /3 3 8 8 4 ( )! 4/5! 4 5 1 1 (! 3) (! 3) 4 (! ) 16 Rev.S10 3 Let s Practice Some Simplification (Cont.) Use positive rational exponents to write each expression. a) 5 4 b) 3 6 x xi x s 5 x 4 ( x 4 ) 1/5 x 4/5 1/ 1/ 3 6 1/3 1/ 6 1/3 + 1/ 6 xi x x ix x ( x ) 1/ 1/ 1/ 4 x ( ) ( ) ( ) ( ) Rev.S10 33 http://faculty.valenciacc.edu/ashaw 11
What are Power Functions? Power functions typically have rational exponents. A special type of power function is a root function. Examples of power functions include: f 1 (x) x, f (x) x 3/4, f 3 (x) x 0.4, and f 4 (x) 3 x Rev.S10 34 What are Power Functions? (Cont.) Often, the domain of a power function f is restricted to nonnegative numbers. Suppose the rational number p/q is written in lowest terms. The the domain of f(x) x p/q is all real numbers whenever q is odd and all nonnegative numbers whenever q is even. The following graphs show 3 common power functions. Rev.S10 35 Example Modeling Wing Size of a Bird: Heavier birds have larger wings with more surface areas than do lighter birds. For some species the relationship can be modeled by S(w) 0.w /3, where w is the weight of the bird in kilograms and S is surface area of the wings in square meters. (Source: C. Pennycuick, Newton Rules Biology.) a) Approximate S(0.75) and interpret the result. b) What weight corresponds to a surface area of 0.45 square meter? Rev.S10 36 http://faculty.valenciacc.edu/ashaw 1
Example (Cont.) a) S(0.75) 0.(0.75) /3! 0.165. The wings of a bird that weighs about 0.75 kilogram have the surface area of about 0.165 square meter. b) To answer this, we must solve the equation 0.w /3 0.45. Rev.S10 37 (cont.) Example (Cont.) /3 0.w 0.45 w /3 /3 3 ( w ) 0.45 0. 3 3 3! 0.45 " # $ % 0. &! 0.45 " w # $ % 0. &! 0.45 " w ± # $ % 0. & w ' ± 3.4 Since w must be positive, the wings of a 3.4 kilogram bird must have a surface area of about 0.45 square meter. Rev.S10 38 How to Solve Equations Involving Rational Exponents? Example Solve 4x 3/ 6 6. Approximate the answer to the nearest hundredth, and give graphical support. s Symbolic 4x 3/ 6 6 4x 3/ 1 (x 3/ ) 3 x 3 9 x 9 1/3 x.08 Graphical Rev.S10 39 http://faculty.valenciacc.edu/ashaw 13
Check How to Solve Equations Involving Rational Exponents? (Cont.) 3 ( 1)! 1! 1 "! 1 ( ) 3 6! 6! 4 4 Substituting these values in the original equation shows that the value of 1 is an extraneous solution because it does not satisfy the given equation. Therefore, the only solution is 6. Rev.S10 40 How to Solve Equations Involving Radicals? Some equations may contain a cube root. Solve 4x! 4x + 1 x. 3 3 4x! 4x + 1 x 3 3 3 3 3 3 ( 4x! 4x + 1) ( x) 4x! 5x + 1 0 ( 4x! 1)( x! 1) 0 1 x or x 1 4 Both solutions check, so the solution set is! 1 " #, 1 $. % 4 & Rev.S10 41 How to Solve Equations Involving Negative Exponents?!! 1 Example Solve 6x + x.!! 1 6x + x 6u + u! 0 ( u )( u ) 3 +! 1 0 1 u! or u 3 x! 3 or x 1 1 Since u, then x. x u Rev.S10 4 http://faculty.valenciacc.edu/ashaw 14
We have learned to What have we learned? 1. identify a rational function and state its domain.. find and interpret vertical asymptotes. 3. find and interpret horizontal asymptotes. 4. solve rational equations. 5. solve applications involving rational equations. 6. solve applications involving variations. 7. solve polynomial inequalities. 8. solve rational inequalities. Rev.S10 43 What have we learned? (Cont.) 9. learn properties of rational exponents. 10. learn radical notation. 11. use power functions to model data. 1. solve equations involving rational exponents. 13. solve equations involving radical expressions Rev.S10 44 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S10 45 http://faculty.valenciacc.edu/ashaw 15