Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics §6.1 Rational FcnMult & Div Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

MTH 55 5.7 Review § • Any QUESTIONS About • §AppG → Graphing Rational Functions • Any QUESTIONS About HomeWork • §AppG → HW-22

Recall Rational Function • A rational function is a function, f(x), that is a quotient of two polynomials; i.e.  • Where • where p(x) and q(x) are polynomials and where q(x) is NOT the ZERO polynomial. • The domain of f consists of all inputs x for which q(x) ≠ 0.

Rational FUNCTION Example • RATIONAL FUNCTION ≡ a function expressed in terms of rational expressions • Example  Find f(3) for this Rational Function: • SOLUTION

Find the Domain of a Rational Fcn • Write an equation that sets the denominator of the rational function equal to 0. • Solve the denominator equation. • Exclude the value(s) found in step 2 from the function’s domain.

Example  Find Domain • Find the Domain for • SOLUTION Set the denominator equal to 0. Factor out the monomial GCF, y. FOIL Factor the 2nd Degree polynomial Use the zero products theorem. • The fcn is undefined for y = 0, −4, or −1, so the domain is {y|y −4, −1, 0}.

Example  Find Domain • Find the Domain for • SOLUTION • Find the values of x for which the denominator x2 – 6x + 8 = 0, then exclude those values from the domain. • The fcn is undefined for x = 2, or 4, so the domain is {x|x 2, 4}. • Interval Notation: (−∞,2)U(2,4)U(4,∞)

Example  Graph • SOLUTION: x 1, so the graph has a vertical asymptote at x = 1. Find ordered pairs around the asymptote and then graph.

Simplifying Rational Expressions and Functions • As in arithmetic, rational expressions are simplified by “removing”, or “Dividing Out”, a factor equal to 1. • example equals 1 removed the factor that equals 1

Maintain Domain • Because rational expressions often appear when we are writing functions, it is important that the function’s domain not be changed as a result ofsimplifying. For example, the Domain of the function given by is assumed to be all real numbers for which the denominator is NONzero

Maintain Domain • Thus for Rational Fcn: • In the previous example, we wrote F(x) in simplified form as • There is a serious problem with stating that these are equivalent; The Domains are NOT the same

Maintain Domain • Why ≠ • The domain of the function given by • Thus the domain of G includes 5, but the domain of F does not. This problem can be addressed by specifying

Example  Maintain Domain • Write this Fcn in Simplified form • SOLUTION: first factor the numerator and denominator, looking for the largest factor common to both. Once the greatest common factor is found, use it to write 1 and simplify as shown on the next slide

Example  Maintain Domain Note that the domain of g = {x| x  2/3 and x −7}by Factoring (see next) Factoring. The greatest common factor is (3x− 2). Rewriting as a product of two rational expressions. For x 2/3, we have (3x −2)/(3x − 2) = 1.

Example  Maintain Domain Removing the factor 1. To keep the same domain, we specify that x 2/3. • Thus the simplified form of

“Canceling” Confusion • The operation of Canceling is a ShortHand for DIVISION between Multiplication Chains • Canceling can ONLY be done when we have PURE MULTIPLICATION CHAINS both ABOVE & BELOW the Division Bar`

Canceling Caveat • “Canceling” is a shortcut often used for removing a factor equal to 1 when working with fractions. Canceling removes multiplying factors equal to 1 in products. It cannot be done in sums or when adding expressions together. Simplifying the expression from the previous example might have been done faster as follows: When a factor that equals 1 is found, it is “canceled” as shown. Removing a factor equal to 1.

Canceling Caveat Caution!Canceling is often performed incorrectly: Incorrect! Incorrect! Incorrect! In each situation, the expressions canceled are not both factors. Factors are parts of products. For example, 5 is not a factor of the numerator 5x – 2. If you can’t factor, you can’t cancel! When in doubt, do NOT cancel! To check that these are not equivalent, substitute a number for x.

Simplifying Rational Expressions • Write the numerator and denominator in factored form. • Divide out all the common factors in the numerator and denominator; i.e., remove factors equal to ONE • Multiply the remaining factors in the numerator and the remaining factors in the denominator.

Example  Simplify • SOLUTION: Factor out the GCF. Factor the polynomial factors. Divide out common factors.

Multiply Rational Expressions • The Product of Two Rational Expressions • To multiply rational expressions, multiply numerators and multiply denominators: • Then factor and simplify the result if possible.

Example  Multiplication • Multiply and, if possible, simplify. a) b) • SOLUTION a) MULTIPLICATIONChains → Canceling OK

Example  Multiplication • SOLUTION b) MULTIPLICATIONChains → Canceling OK

Example  Multiply & Simplify • Multiply and,if possible, simplify. • SOLUTION MULTIPLICATIONChains → Canceling OK

Divide Rational Expressions • The Quotient of Two Rational Expressions • To divide by a rational expression, multiply by its reciprocal • Then factor and, if possible, simplify.

Example  Division • Divide and, if possible, simplify. a) b) • SOLUTION a) Multiplying by the reciprocal of the divisor Multiplying rational expressions b)

Example  Division • Divide and, if possible, simplify. • SOLN MULTIPLICATIONChains → Canceling OK

Example  Division • Divide and, if possible, simplify.  • SOLUTION MULTIPLICATIONChains → Canceling OK

Example  Division • Divide and, if possible, simplify.  MULTIPLICATIONChains → Canceling OK • SOLUTION

Example  Manufacturing Engr • The function given by • gives the time, in hours, for two machines, working together to complete a job that the 1st machine could do alone in t hours and the 2nd machine could do in 3t − 2 hours. • How long will the two machines, working together, require for the job if the first machine alone would take (a) 2 hours? (b) 5 hours?

Example  Manufacturing Engr • SOLUTION (a) (b)

WhiteBoard Work • Problems From §6.1 Exercise Set • 22 (ppt), 26 (ppt), 114 , 16, 46, 66, 86 • More Rational Division • Since we are dividing fractions, we multiply by the reciprocal • Now, we follow the rule for multiplication • Factor and then cancel • Don't leave the numerator empty - put a one to hold the place.

P6.1-22  Rational fcn Graph • Describe end-behavior of Graph at Far-Right • ANS: As x becomes large y = f(x) approaches, but never reaches, a value of 3

P6.1-22  Rational fcn Graph • What is the Eqn for the Horizontal Asymptote: • ANS: y = f(x) approaches, but never reaches, a value of 3, to the Asymptote eqn y = 3

P6.1-26  Rational fcn Graph • List 2 real No.s that are NOT function values of f • ANS: y = f(x) does not have a graph between y > 0 and y < 3. Thus two values for which there is NO f(x): • y = 1 or y = 2

P6.1-114  Smoking Diseases • Find P(9). Describe Meaning. ID pt on Graph • ANS: An incidence ratio of 9 indicates that 88.9% of Lung Cancer deaths are associated with Cigarette smoking

P6.1-114  Smoking Diseases • ID P(9) on Graph

All Done for Today • The FIRST Algebraist • In the 3rd century, the Greek mathematician Diophantus of Alexandria wrote his book Arithmetica. Of the 13 parts originally written, only six still survive, but they provide the earliest record of an attempt to use symbols to represent unknown quantities. Diophantus ofAlexandria

Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

Graph y = |x| • Make T-table

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege