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Section 6.1 Rational Expression & Functions: Definitions, Multiplying, Dividing

Section 6.1 Rational Expression & Functions: Definitions, Multiplying, Dividing. Fractions - a Quick Review Definitions : Rational Functions, Expressions Finding the Domains (and Exclusions) of Rational Functions Simplifying Rational Functions Simplifying by factoring out -1.

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Section 6.1 Rational Expression & Functions: Definitions, Multiplying, Dividing

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  1. Section 6.1 Rational Expression & Functions:Definitions, Multiplying, Dividing • Fractions - a Quick Review • Definitions: Rational Functions, Expressions • Finding the Domains (and Exclusions) of Rational Functions • Simplifying Rational Functions • Simplifying by factoring out -1 6.1

  2. Fractions - Review • Q: When can you add or subtract fractions? • A: Only when denominators are the same • Q: What do you do when denominators are not the same? • A: Use their LCD to create equivalent fractions. • Q: How do you multiply fractions? • A: Factor all tops and factor all bottoms, cancel matching factors, multiply tops and bottoms • Q: What do you do first when dividing fractions? • A: Turn division into multiplication : reciprocal the divisor. • Rational Expressions are Polynomial Fractions ! Same rules! 6.1

  3. Definitions 6.1

  4. Finding the Domain (and exclusions) of a Rational Function Recall the domain of a function is the set of all real numbers for which the function is defined. • What real values make this function undefined (divided by 0)? Factor: x2 + 2x – 24 = (x – 4)(x + 6) {x | x is Real, except for4 or -6} 6.1

  5. Graphs of Rational Functions t=-5/2 is an Asymptote 2t + 5 ≠ 0 2t ≠ -5 t ≠ -5/2 6.1

  6. Definitions • Horizontal Asymptote – A horizontal line that the graph of a function approaches as x values get very large or very small. • Vertical Asymptote – A vertical line that the graph of a function approaches as x values approach a fixed number 6.1

  7. More Properties of Fractions - Review 6.1

  8. Simplifying Rational Expressions(In general, the expressions are NOT equivalent) 6.1

  9. First Factor and Identify domain exclusions,Then Simplify 6.1

  10. Multiplying Fractions (First find domain exclusions) Factor expressions, then cancel like factors 6.1

  11. Example – Step by Step x≠0,3 x 1 1 4 1 1 Write down original problem Combine with parentheses Find any polynomials that need factoring Rewrite (if any factoring was done) Identify domain exclusions Cancel out matching factors Simplify the answer 6.1

  12. Board Practice – Rational Multiplication • Write original problem • Combine w/ parens • Factor polynomials • Rewrite (if any factoring) • Identify domain exclusions • Cancel matching factors • Simplify the answer 6.1

  13. Finding Powers of Rational Expressions • Factor and Simplify (if possible) before applying the power • If part of a larger expression, see if any terms cancel out • Multiply out the terms in the numerator,multiply out the terms in the denominator. • Leave in simplified factored form 6.1

  14. Dividing Fractions Change Divide to Multiply by Reciprocal,followmultiply procedure 6.1

  15. Board Practice- Rational Division • Write original problem • Combine w/ parens • Factor polynomials • Rewrite (if any factoring) • Identify domain exclusions • Cancel matching factors • Simplify the answer 6.1

  16. Mixed Operations • Multiplications & Division are done left to right • In effect, make each divisor into a reciprocal 6.1

  17. What Next? • Present Section 6.2 Add/Subtract Rational Expressions 6.1

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