12.6 What is a rational function?

What is a rational function?

Lessons

Notes:
A rational function is defined as a "ratio" of polynomials:
rationalfunction=polynomialpolynomialrational\;function = \frac{{polynomial}}{{polynomial}}
For example: f(x)=x3+5x28x+6x21f\left( x \right) = \frac{{{x^3} + 5{x^2} - 8x + 6}}{{{x^2} - 1}} ; g(x)=1x24g\left( x \right) = \frac{1}{{{x^2} - 4}} ; h(x)=8x+32x5h\left( x \right) = \frac{{ - 8x + 3}}{{2x - 5}}
    • a)
      Complete the table of values below, then plot the points on the grid.

      xx

      -5

      -4

      -3

      -2

      -1

      0

      1

      2

      3

      4

      5

      y=f(x)=1x2y = f\left( x \right) = \frac{1}{{x - 2}}

    • b)
      What is the non-permissible value of the rational function?
    • c)
      c) Now, let’s investigate the behaviour of the rational function near the non-permissible value by plotting more points close to the non-permissible value.

      xx

      1.5

      1.9

      1.99

      2

      2.01

      2.1

      2.5

      y=f(x)=1x2y = f\left( x \right) = \frac{1}{{x - 2}}

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    • d)
      To investigate the right-end behaviour of the rational function (as xx \to \infty ), complete the table of values below and plot the points.

      xx

      10

      100

      1000

      y=f(x)=1x2y = f\left( x \right) = \frac{1}{{x - 2}}

    • e)
      e) To investigate the left-end behaviour of the rational function (as xx \to - \infty ), complete the table of values below and plot the points.

      xx

      -10

      -100

      -1000

      y=f(x)=1x2y = f\left( x \right) = \frac{1}{{x - 2}}

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What is a rational function?

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