Rational Functions Math Example 1

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Example 1

medium
Find the vertical and horizontal asymptotes of f(x)=3x+2x−4f(x) = \frac{3x + 2}{x - 4}.

Solution

  1. 1
    Vertical asymptote: set denominator =0= 0: x−4=0x - 4 = 0, so x=4x = 4.
  2. 2
    Horizontal asymptote: numerator and denominator have the same degree (both degree 1).
  3. 3
    The horizontal asymptote is y=leading coefficient of numeratorleading coefficient of denominator=31=3y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{3}{1} = 3.

Answer

Vertical: x=4,Horizontal: y=3\text{Vertical: } x = 4, \quad \text{Horizontal: } y = 3
For rational functions, vertical asymptotes occur where the denominator is zero (and the numerator is not). The horizontal asymptote depends on comparing the degrees of numerator and denominator.

About Rational Functions

A rational function is a ratio of two polynomials: f(x)=P(x)/Q(x)f(x) = P(x)/Q(x) where PP and QQ are polynomials and Q(x)≠0Q(x) \neq 0.

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More Rational Functions Examples

Example 2 hard

Find all asymptotes and holes of [formula].

Example 3 medium

Find the [formula]-intercepts and vertical asymptotes of [formula].

Example 4 hard

Find all asymptotes (vertical, horizontal, and oblique) of [formula]. Does the function have a hole

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