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Chapter 5 Rational Numbers and Rational Functions

You will have to read chapter 4 to better understand some of the terminology in this chapter Rational numbers are closed with respect to the basic operations (arithmetic, addition and multiplication) i.e. the sum or product of any two rational numbers is rational

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Chapter 5 Rational Numbers and Rational Functions

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  1. You will have to read chapter 4 to better understand some of the terminology in this chapter • Rational numbers are closed with respect to the basic operations (arithmetic, addition and multiplication) • i.e. the sum or product of any two rational numbers is rational • Open your book to page 288 (we are going to look at Example 1) Chapter 5 Rational Numbers and Rational Functions Section 5-1 Rational Numbers and Rational Expressions

  2. You must always check for restrictions to the solution (numbers that cannot be used) • When simplifying fractions, you can eliminate identical factors but not identical terms (see page 290) • Ex1. Simplify by factoring • Ex2. Simplify • Ex3. Multiply Never leave answers in complex form!

  3. Section 5-2 Irrational Numbers • Open your book to pages 295-296 where we will look at the theorem and proof about irrationality • Familiarize yourself with the Law of Indirect Reasoning (modus tollens) • There are more irrational numbers than rational numbers • Proof by contradiction is often used to prove properties involving irrational numbers (see example 1 on page 297) • The negation symbol is ~

  4. Do not leave radicals in the denominator • If there is an expression that contains a radical, multiply by the conjugate • Ex1. Rationalize the denominator • Ex2. Rationalize the denominator • Ex3. Rationalize the denominator

  5. Section 5-3 Reciprocals of Power Functions • The behavior of a hyperbola near x = 0 can be used to analyze many other functions and their graphs • Graph a hyperbola on your calculator • As the graph approaches 0 from the right, the function gets larger • This is written: • As the graph approaches 0 from the left, the functions gets smaller. Written:

  6. So the + and – symbols as exponents in the limit are directional symbols (from the left or right) • Ex1. Consider the graph of the function f with rule where x ≠ 3 a) Determine an equation for the vertical asymptote b) Use limit notation to describe the behavior of the function near the vertical asymptote

  7. Section 5-4 Rational Functions • All polynomial functions are rational functions • The sum, difference, product and quotient of two rational functions are rational functions • A function f is said to have a removable discontinuity at a point x if the graph of f has a hole at x but is otherwise continuous on an interval around x (see graphs on page 308-309) • The term removable means that the function could be made continuous by redefining its value at that one point

  8. Discontinuity that cannot be removed by insertion of a single point is called an essential discontinuity • Ex1. a) Determine the location(s) of the removable discontinuities of the function f with b) How can these discontinuities be removed? • Ex2. Show that has essential discontinuities at x = -2 and at x = 3. • A nonrational function may have an essential discontinuity without an asymptote (i.e. the floor function)

  9. Section 5-5 End Behavior of Rational Functions • The end behavior of any rational function is the same as that of one of three functions: a power function, a constant function, or the reciprocal of a power function • The end behavior of a function f is the behavior f(x) as │x│→∞ • Determine the end behavior of any polynomial function by examining its degree and the sign of its leading coefficient

  10. When you are asked to describe the end behavior of a function, use limits • Open your book to page 315 (example 1) • When a rational function is written as a quotient of polynomials, its end behavior is found by dividing the leading terms of the polynomials (see theorem on page 317) • Ex1. Describe the end behavior of the function f defined by the formula • Ex2. Let h be the function defined by a) Rewrite h using long division b) Describe the behavior of h as x→+∞ and as x→-∞

  11. Section 5-6 The Tan, Cot, Sec, and Csc Functions • You can create a rational function by using the reciprocal of a polynomial function or by dividing one polynomial function by another • These four functions are NOT rational functions • This section is review from the Trig part of the year • You might want to familiarize yourself with the graphs of these functions (see pages 322 and 323)

  12. Section 5-7 Rational Equations • Multiplying both sides of an equation by an expression which contains a variable is NOT a reversible step when the value of the expression is zero • For this reason, after you solve you must check that all of the obtained values are actually solutions • Ex1. Solve • Open your book to page 328 (example 3)

  13. Ex2. Consider the situation of example 3 from the book. If upstream flooding increased the current’s rate by 2mph, what would the still water speed of the boat have to be to make the same round trip in 15 minutes?

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