8.1 Simplify Rational Functions (Page 1 of 25) Chapter 8 Rational Functions 8.1 Simplifying Rational Expressions; Finding the Domain and Range of Rational Functions Rational Function If P(x) and Q(x) are polynomials, then a rational function f is a function that can be written in the form f (x) P(x) , where Q(x) 0 . Q(x) Some examples of rational functions are x 3 3x 6 2x 2 17 f (x) , g(x) , x8 5x 1 h(x) 4 . 5x 3 Domain of a Rational Function The domain of a rational function is the set of all real numbers except where Q(x) 0 . Example 1 Find the domain of x 3 3x 2x 2 17 a. f (x) b. g(x) x8 5x 1 c. h(x) 4 5x 3 8.1 Simplify Rational Functions (Page 2 of 25) Vertical Asymptote A vertical asymptote is a vertical line that the graph of a function gets arbitrarily close to, but never touches nor crosses. For example, the graph of f in figure 1 has a vertical asymptote at x = 5. A Vertical asymptotes can only occur at a value of x that make the denominator zero. f ( x) 4 4 –4 x 5 Figure 1 Graph of f (x ) Let g(x) QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 2. The sketch of g is shown in figure 2. Write the equation of each vertical asymptote on the graph of g. TI-83/84 ZStandard 3. What is the equation of the horizontal asymptote? 4. What is the range of g? Figure 2 Graph of g(x ) x5 8 Example 2 6x x 2 25 1. Find the domain of g. 3 6x (x 25) 2 3 x 5 8.1 Simplify Rational Functions (Page 3 of 25) Example 3 The graph g(x) shown. x4 is 3x 2 7x 10 1. Find the domain of g. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 2. Find the equation of any vertical asymptote in the graph of g. 3. Find the y-intercept in the graph of g. TI-83/84 ZStandard 8.1 Simplify Rational Functions (Page 4 of 25) A Simplified Rational Expression A rational expression is simplified if the numerator and denominator have no common factors other than 1. To Simplify a Rational Expression 1. Factor the numerator and denominator. 2. Divide out any common factors. Example 4 x 2 x 12 Simplify the expression 3 x 4 x 2 5x 20 Example 5 1. Simplify ab ba 2. Simplify x5 5 x 3. Simplify 2x 7 7 2x 4. Simplify 11x 3 3 11x 8.1 Simplify Rational Functions (Page 5 of 25) Example 6 Find the domain of each function and simplify the righthand side of the equation. Check your result with the graphing and table functions of you calculator. 2x 2 6x 20 f (x) 1. 2x 2 50 2. x 3 5x 2 6x g(x) 3 x 2x 2 9x 18 8.1 Simplify Rational Functions (Page 6 of 25) Example 7 Let f (x) 8x 3 125 and g(x) 4x 2 25 . f 1. Find the equation for (x) and simplify the right g hand side of the equation. f 2. Find (3) . g 8.2 Multiply and Divides Rational Expressions (Page 7 of 25) 8.2 Multiply and Divide Rational Expressions The following properties assume division by zero does not occur. A C AC Multiplying Rational Expressions B D BD Dividing Rational Expressions A C A D AD B D B C BC Example 1 Perform the indicated operation and simplify. 4 x 3 3x 5 1. 7x 1 2x 2. x2 9 4x 2 25 2x 2 x 10 x 2 4x 21 8.2 Multiply and Divides Rational Expressions (Page 8 of 25) 3. 6x 2 4 x7 x 3 x 1 4. x 2 2x 48 3x 2 9x 6x 24 x 2 8x 16 x 2 16 5x 30 8.2 Multiply and Divides Rational Expressions (Page 9 of 25) Example 2 35x 2 25x 6 x Let f (x) and . g(x) 2 4 x 36 15x 1. Find an equation of the product function f g . 2. Find ( f g)(2) . Example 3 81x 2 49 7 9x Let f (x) 2 and g(x) . 3x 16x 5 18x 6 1. Find an equation of the quotient function 2. Find (3) . g f f . g 8.3 Add and Subtract Rational Expressions (Page 10 of 25) 8.3 Add and Subtract Rational Expressions Let A, B and C be rational expressions and C 0 . Then A B A B Adding Rational Expressions C C C Subtracting Rational Expressions A B A B C C C Example 1 Perform the indicated operation and simplify. x 2 5x 6 1. x2 9 x2 9 2. x2 2x 8 x 2 7x 10 x 2 7x 10 8.3 Add and Subtract Rational Expressions (Page 11 of 25) Example 2 Perform the indicated operation and simplify. x 5 3. x 2 25 x 2 25 4. 3x 2 5x 2x 2 7x 15 2 2 x 10x 21 x 10x 21 Least Common Denominator (LCD) The least common multiple (LCM) of two polynomials is the simplest polynomial that has both polynomials as factors. The least common denominator (LCD) of two rational expressions is the LCM of the denominators. To Find the LCD of Two Rational Expressions 1. Factor each denominator. 2. The LCD is the product of each distinct prime factor the greatest number of time it occurs in a single denominator. 8.3 Add and Subtract Rational Expressions (Page 12 of 25) Example 3 Perform the indicated operation and simplify. 7 6 7 6 1. 3 3 3x 2x 3x 2x 2. 3 5 x5 x3 3. 3 x x 2 9 2x 6 8.3 Add and Subtract Rational Expressions (Page 13 of 25) Example 4 Perform the indicated operation and simplify. 3x 1 5 1. 2x 2 7x 4 x 2 8x 16 2. x 3 x2 2 x 3. 6 3 x2 2 x x x 2 1 x 2 x 8.3 Add and Subtract Rational Expressions (Page 14 of 25) Example 5 x 1 x 1 Let f (x) and g(x) . x 1 x 1 1. Find an equation of the difference function f g . 2. Find ( f g)(2) . Example 6 3 x 1 and . g(x) 3 2 2 12x 22x 6x 30x 10x Find an equation of the sum function f g . Find ( f g)(2) . Let f (x) 1. 2. 8.5 Solve Rational Equations (Page 15 of 25) 8.4 Simplify Complex Rational Expressions Complex Rational Expression A complex rational expression is a rational expression whose numerator and/or denominator is also a rational expression. Some examples of complex rational expressions are x2 9 3 1 5 2 2 x 2 2x 1 , 2x x , x2 2x 6 2 1 x x 1 2 4x 4 5 x x 2 x 4x 12 Simplify Complex Rational Expressions – Method 1 1. Simplify the numerator and denominator of the complex rational expression so that each only have one rational expression. 2. Rewrite the division as multiplication A B A C A D AD C B D B C BC D 3. Factor the numerator and denominator and simplify. Example 1 12 Simplify x 8 x3 8.5 Solve Rational Equations (Page 16 of 25) Example 2 x2 9 2 x 2x 1 Simplify 2x 6 4x 4 Example 3 3 1 2 Simplify 2x x 2 1 5 x 8.5 Solve Rational Equations (Page 17 of 25) Simplify Complex Rational Expressions – Method 2 1. Create a simple rational expression by multiplying the numerator and denominator of the primary rational expression by the LCD of the “inside rational expressions.” 2. Factor the numerator and denominator and divide out any common factors. Example 2 1. 12 Simplify x 8 x3 2. x2 9 2 x 2x 1 Simplify 2x 6 4x 4 8.5 Solve Rational Equations (Page 18 of 25) Example 3 Example 4 Let f (x) 2 3 1 2 Simplify 2x x 2 1 5 x 5 x x 1 and g(x) . 2 x2 x 2 x 4x 12 1. Find an equation of the quotient function 2. Find (4) . g f f . g 8.5 Solve Rational Equations (Page 19 of 25) 8.5 Solve Rational Equations Rational Equation A rational equation is two equal rational expressions. Steps to Solve a Rational Equation 1. Clear all denominators (i.e. make all denominators 1) in the equation by multiplying both sides of the equation by the LCD of all rational expressions. 2. Solve the simplified equation from step 1. 3. Check your solution(s). This step is particularly important with rational equations because it is possible to get what is called an extraneous solution – a false solution that forces division by zero in the original equation. Example 1 3x 1 2 7 Solve . 4x 3x 6x 8.5 Solve Rational Equations (Page 20 of 25) Example 2 Solve for x. 1 3 x 2 x 2 Example 3 Solve for x. x 1 3 2 x Example 4 Solve for x. 2x 4 1 x2 x2 8.5 Solve Rational Equations (Page 21 of 25) Example 5 Solve for x. 3x 4 2 x8 x8 Example 6 Solve for x: x 7 14 . 2 x 2 5 x x 3x 10 8.5 Solve Rational Equations (Page 22 of 25) Example 5 x 1 x 2 Let f (x) . x3 x3 a. Without using the graphing capability of your calculator, find the exact value of x when f (x) 1 . Then round those values to 2 decimal places. b. Verify your results graphically. That is, set Y1 f (x) and Y2 1. Then intersect the two graphs on your calculator. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 8.6 Modeling with Rational Functions (Page 23 of 25) 8.6 Modeling with Rational Functions Example 1 The ski club plans to spend $1250 to charter a bus for a ski trip. Each participant will pay an equal share of the bus cost plus $350 for food, lodging and lift tickets. 1. Let C(n) represent the total cost (in dollars) for n participants. Find the equation for C. 2. Let the function M(n) represents the mean cost per student. Find the equation for M. What are the units of M? 3. What is M(20)? Explain its meaning in this application. 4. What is the minimum number of participants required so that each person’s cost is no more than $400? 8.6 Modeling with Rational Functions (Page 24 of 25) Example 2 College textbook sales (in millions of dollars) for various years are given in the first table. The number (in millions) of men and women enrolled in U.S. colleges for various years is given in the second table. 1. Let C(t) represent the total cost of textbooks (in millions of dollars) for t years since 1980. Find the function C. Year 1980 1985 1990 1995 2000 Year 1980 1985 1990 1995 2000 Textbook Sales (millions of dollars) 1801 2410 3445 4310 4866 Enrollment (millions) Women Men 6.0 5.4 6.6 5.9 7.4 6.2 8.0 6.7 8.7 7.1 2. Let E(t) represent the total enrollment in U.S. colleges for t years since 1980. Find the function E. 3. Let M(t) represent the mean amount of money spent on textbooks per student. Find the function M. 4. Predict the mean amount of money that each student will spend on textbooks in 2005. 5. When will the average student have to spend $500 on textbooks. 8.6 Modeling with Rational Functions (Page 25 of 25) Example 3 Suppose a student plans to drive from Miami, Florida to Atlanta, Georgia. The speed limit is 70 mph in Florida and 65 mph in Georgia. The trip involves 473 miles in Florida d an 253 miles in Georgia. Recall that since r t d , t . r 1. If the student drives the speed limit, how much driving time does it take? 2. Forget driving the speed limit - it takes too long. Find the function T (a) that represents the time it takes the student to complete the trip driving a mph above the speed limit. 3. Find T (0) and explain its meaning. 4. Find T (8) and explain its meaning. 5. If the student wants the driving time to be 9 hours, how much over the speed limits must she drive?