6'1: Rational Fcns and Eqns

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6.1 Rational Fcns and Eqns

• Recall a rational number is one that can be
  written in the form a/b
• Rational Number?
• 2/3
• Yes
• 2 1/2
• Yes 5/2
• 3.6
• Yes 18/5

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Rational Number?

• 6
• Yes 6/1
• Pi
• No! a non-terminating, non-repeating decimal
• Sqrt(16)
• Yes 4 4/1
• Sqrt(7)
• No! a non-terminating, non-repeating decimal

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Polynomials

• Recall Polynomials
• Monomial
• 2x, 4, 3x2
• Bimonial
• 2x 4 4 - x 3x2 - 2x
• Trinomial
• 3x2 - 2x 3 5x5 - 2x3 2
• Polynomial
• 4 or more terms

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Rational Expressions

• Are formed when a polynomial is divided by a
  non-zero polynomial
• Need an x term in the denominator after
  simplifying.
• Recall n/0 is an undefined operation.
• Some examples

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Rational Expression?

• Yes! Both numerator and denominator are
  polynomials!
• No! Denominator is not a polynomial sqrt(x)

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Rational Expression?

• Yes both N D are polynomials!
• No! Denominator not a polynomial!

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Rational Function

• Let p(x) and q(x) be polynomials. Then a
  rational function is given by

The domain (values x can take) of the fcn
includes all x-values such that q(x) ? 0.
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State the domain

• The domain is all values of x, except the
  denominator ? 0
• Solve x - 1 ? 0
• x ? 1

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State the domain

• The domain of f(x) is all values of x, but 2x - 4
  ? 0.
• Solve the equation 2x - 4 ? 0
• x ? 2

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State the domain

• The domain of f is all values of x, but x2 - 4
  ? 0 solve by factoring
• (x - 2)(x 2) ? 0
• By ZPP, x-2 ? 0 x2 ? 0
• x ? ? 2

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State the domain

• Use SRP to solve x2 1 ? 0
• x2 ? -1
• x ? ? sqrt(-1)
• No real solutions . There are no restrictions on
  the values x can take.
• See graph!

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Evaluating a rational function

• Ex. 3 A rational fcn is represented numerically,
  symbolically, and graphically. Use each to
  evaluate f(-1), f(1), and f(2).

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Numerically f(-1), f(1), f(2)

• f(-1) 1
• f(1) Error undefined
• f(2) 4

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Symbolically find f(-1)
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Symbolically find f(1)
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Symbolically find f(2)
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Graphically find f(-1)
f(-1) 1, When x -1, the y-value of the graph
is 1
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Graphically find f(2)
f(2) 4 When x 2, the y-value of the graph is 4.
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Graphically find f(1)
f(1) ?
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Asymptote

• Since the fcn weve been dealing with is
  undefined when x 1, the graph of the fcn will
  never cross or touch the vertical line x 1.
  This is the vertical asymptote.

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Sketch the vertical asymptote

• y 2/(2x - 1)
• y x2/(x2)
• y 4x/(x2 - 9)
• y 3x/(3x - 2)
• y 5x2/(x2 - 5)

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Solving Rational Eqns

• Recall To clear fractions out of an equation
  multiply both sides by the LCD.
• Check your answer!!!
• REAL IMPORTANT to check answers of a rational
  equation to make sure an answer doesnt give us a
  zero in the denominator.

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Solve

• MBS by (2x-1)
• 3x 6x - 3
• Now solve for x
• x 1
• Check answer in original eqn

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Solve

• MBS by (x-2) .
• x 1 3
• Solve for x
• x 2
• Check by substituting into original eqn..
• We get zero in the denominator .. An undefined
  operation thus x 2 is not a valid solution.

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Solve

• X -2
• X 1/2

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Rational Eqns in the TI-83
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Homework 6.1

• 2-5 7-12 19-22 24-26 29-34 37-47 odds
  61-64, 78.