Rational Functions

Notes:

Terms

Rational Function is a function of the form:
R(x) = n(x)/d(x)
where n(x) and d(x) are polynomials.

Domain: Input ‘range’ of x.

Range: Output ‘range’ of y.

Y-axis: “Vertical Asymptote”

  • Vertical Asymptote:
    A Vertical Asymptote for a rational function is a line x=k;
    Where ‘k’ is a zero of the denominator of the function.

Example:

f(x) = (2x+5)/(x-2)

When x is close to 2, x-2 is close to 0, so dividing by a very small number, which gives us large numbers.

”(1 - 2), (1.5 - 2), (1.75 - 2), (1.9 - 2), (1.999999 - 2)”

X-axis: “Horizontal Asymptote”

  • Horizontal Asymptote:
    A Horizontal line ‘y=k’ such that as x gets larger, R(x) gets closer to ‘k’.
    In other works… “As x -> ∞; R(x) -> k”

Diagonal Asymptote:

Example:
Given function… f(x) = (x2 + 2x - 8)/(x-1)
We can divide simplify the expression by completing the division; Which will give us the equation (with a remainder): (x + 3) - (5)/(x-1).

The Diagonal Asymptote is then defined by the equation: y = x + 3

Summary of Vertical and Horizontal asymptotes:

Vertical asymptotes are lines ‘x=k’; Where k is a zero of d(x) but not of n(x).

  • Vertical asymptotes correspond to zeros of the denominator!

Extra rules:

If (n(x) has a smaller degree than d(x)) {
  then 'y = 0' (the x-axis) is a Horizontal Asymptote;
}
Else if (n(x) and d(x) have the same degree) {
  then 'y = (a/b)' is the Horizontal Asymptote {
    Where 'a' and 'b' are the leading coefficients of n(x) and d(x);
  }
}
Else if (n(x) has a higher degree than d(x)) {
  then the Horizontal Asymptote is actually a DIAGONAL Asymptote;
}