A rational function is the quotient of two polynomials'

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2.4 Rational Functions

• A rational function is the quotient of two
  polynomials.

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Objective

• The objective in this section is learn to sketch
  the graph of rational functions.
• To do that, we have to determine
• The domain.
• The axis intercepts.
• The asymptotes.
• The signs.

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The Domain

• The domain of f(x) is the set of all real
  numbers x with Q(x) ? 0

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The Axis Intercepts

• The x-axis intercepts.
• Set f(x) 0 and solve the equation for x.
• The y-axis intercepts.
• Set x 0 and f(x) y and solve the equation
  for y.

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Asymptote

• An asymptote of a graph is a line that the graph
  approaches.

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• Vertical Asymptotes
• Let f(x) a rational function, then a vertical
  asymptote occur at x a if x - a is a
  factor in the denominator of its simplest form,
  that is, after canceled the common factors.

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• Horizontal Asymptotes
• A rational function f(x) can have at most one
  horizontal asymptote, since f(x) approaches a
  finite value as x approaches ?, it approaches
  that same value as x approaches - ?.
• To find the horizontal asymptotes, identify the
  degree of the function (the greater exponent),
  let n, and divide the numerator and denominator
  by xn. Then make x approaches ?.
• Note A rational function will not have a
  horizontal asymptotes when the degree of the
  numerator is greater that the degree of the
  denominator.

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• Slant or Oblique Asymptotes.
• Let f(x) P(x) / Q(x)
• When the degree of the P(x) is just 1 more than
  the degree of the Q(x), the graph approaches a
  non horizontal line as x approaches ? and as x
  approaches -?. This line is called a slant or
  oblique asymptote.
• To find the slant or oblique asymptote, perform
  P(x) / Q(x)

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Problems

• 1. Solve the problem 18, page 164
• 2. Solve the problem 24, page 165
• 3. Solve the problem 36, page 165

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