Rational Functions Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Find all asymptotes and holes of f(x)=x2βˆ’9x2βˆ’xβˆ’6f(x) = \frac{x^2 - 9}{x^2 - x - 6}.

Solution

  1. 1
    Factor numerator: x2βˆ’9=(xβˆ’3)(x+3)x^2 - 9 = (x - 3)(x + 3).
  2. 2
    Factor denominator: x2βˆ’xβˆ’6=(xβˆ’3)(x+2)x^2 - x - 6 = (x - 3)(x + 2).
  3. 3
    Cancel common factor (xβˆ’3)(x - 3): f(x)=x+3x+2f(x) = \frac{x + 3}{x + 2} for xβ‰ 3x \neq 3. There is a hole at x=3x = 3.
  4. 4
    Vertical asymptote: x+2=0β‡’x=βˆ’2x + 2 = 0 \Rightarrow x = -2.
  5. 5
    Horizontal asymptote: degrees are equal, y=11=1y = \frac{1}{1} = 1. Hole yy-value: f(3)=65f(3) = \frac{6}{5}.

Answer

HoleΒ atΒ (3,65),VA:Β x=βˆ’2,HA:Β y=1\text{Hole at } \left(3, \frac{6}{5}\right), \quad \text{VA: } x = -2, \quad \text{HA: } y = 1
When a factor cancels between numerator and denominator, the function has a removable discontinuity (hole) rather than a vertical asymptote at that xx-value.

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)β‰ 0Q(x) \neq 0.

Learn more about Rational Functions β†’

More Rational Functions Examples

Example 1 medium

Find the vertical and horizontal asymptotes 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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