What is a rational function?

What is a rational function?

Lessons

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}}
  • 1.
    Investigating Asymptotes on the Graph of Rational Functions
    Consider the rational function f(x)=1x2f\left( x \right) = \frac{1}{{x - 2}} .
    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)
    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)
    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}}