Rational Functions and End Behavior

Rational Functions and End Behavior

Definition of Rational Functions

  • A rational function is defined as the quotient (fraction) of two polynomials.

    • General form:
      y=f(x)g(x)y = \frac{f(x)}{g(x)}
      where:

    • $f(x)$ and $g(x)$ are both polynomials

    • $g(x)
      eq 0$

Examples of Rational Functions

  1. y=2x+3y = \frac{2}{x+3}

  2. y=x2−3x+13x+4y = \frac{x^2 - 3x + 1}{3x + 4}

  3. y=2x2+4x−6x3−7x+11y = \frac{2x^2 + 4x - 6}{x^3 - 7x + 11}

End Behavior for Rational Functions

  • The end behavior of a rational function is determined by the leading terms of the numerator and the denominator:

    • Let $f(x) = \frac{ax^n}{bx^d}$, where:

    • $n$ is the degree of the numerator

    • $d$ is the degree of the denominator

Case Analysis of End Behavior

  • Case I: Leading terms have the same degree ($n = d$)

    • Result: Function has a horizontal asymptote:
      y=aby = \frac{a}{b} (ratio of the leading coefficients)

  • Case II: Denominator dominates the numerator ($n < d$)

    • Result: Function approaches a horizontal asymptote:
      y=0y = 0

  • Case III: Numerator dominates the denominator ($n > d$)

    • Result: Function behaves like the polynomial:
      y=axn−dy = ax^{n - d}

    • Additional Note: If the degree of the numerator is exactly one more than the degree of the denominator, then the function has a slant (oblique) asymptote.

Examples of Asymptotes

Example 1: Identifying Asymptotes for Rational Functions

  • Determine if the following rational functions have a horizontal asymptote, a slant asymptote, or neither. If a horizontal asymptote exists, write the equation.

    • a) f(x)=3x2+4x−75x2−3f(x) = \frac{3x^2 + 4x - 7}{5x^2 - 3}

    • Result: Horizontal Asymptote at y=35y = \frac{3}{5}

    • b) y=x2+3x+2y = x^2 + 3x + 2

    • Result: No Horizontal Asymptote

    • c) g(x)=2x2−45x+9g(x) = \frac{2x^2 - 4}{5x + 9}

    • Result: Slant Asymptote at $4x + 5$

    • d) y=8x−13y = \frac{8x - 1}{3}

    • Result: No Horizontal Asymptote

    • e) k(x)=3x2+3x−7k(x) = \frac{3}{x^2 + 3x - 7}

    • Result: Horizontal Asymptote at y=0y = 0

    • f) p(x)=2x+15x2p(x) = \frac{2x + 1}{5x^2}

    • Result: Horizontal Asymptote at y=0y = 0

Example 2: Limit Statements for End Behavior

  • Write limit statements to describe the end behavior of the following rational functions:

    • a) f(x)=2x+4x−18x2f(x) = \frac{2x + 4x - 1}{8x^2}

    • Left Limit: lim⁡x→−∞f(x)=−2\lim_{x \to -\infty} f(x) = -2

    • Right Limit: lim⁡x→∞f(x)=13\lim_{x \to \infty} f(x) = \frac{1}{3}

    • b) g(x)=5x2−8x+9g(x) = 5x^2 - 8x + 9

    • Left Limit: lim⁡x→−∞g(x)=0\lim_{x \to -\infty} g(x) = 0

    • Right Limit: lim⁡x→∞g(x)=0\lim_{x \to \infty} g(x) = 0

    • c) h(x)=−3x−x2+xx3+4x+4h(x) = \frac{-3x - x^2 + x}{x^3 + 4x + 4}

    • Left Limit: lim⁡x→−∞h(x)=−1\lim_{x \to -\infty} h(x) = -1

    • Right Limit: lim⁡x→∞h(x)=−1\lim_{x \to \infty} h(x) = -1

Slant Asymptotes

  • Slant asymptotes occur when the degree of the numerator is exactly 1 greater than the degree of the denominator.

  • The slant asymptote is determined as follows:

    • For a rational function of the form: f(x)=axn+ext…bxd+ext…f(x) = \frac{ax^n + ext{…}}{bxd + ext{…}} where:

    • The leading term is $ax^n$ and $bxd$, with $n = d + 1$.

    • It has a slant asymptote parallel to the line:
      y=aby = \frac{a}{b}

Example 3: Identifying Functions with Specific Slant Asymptotes

  • Determine which of the following functions have a slant asymptote parallel to the line y=xy = x:

    • I. f(x)=2x+3x2f(x) = \frac{2x + 3}{x^2}

    • Result: Horizontal Asymptote at y=1y = 1

    • II. g(x)=x2+4x+12x3+x2+2g(x) = \frac{x^2 + 4x + 1}{2x^3 + x^2 + 2}

    • Result: Horizontal Asymptote at y=0y = 0

    • III. h(x)=x2−4xx2+3x+5h(x) = \frac{x^2 - 4x}{x^2 + 3x + 5}

    • Result: Slant Asymptote exists.

    • IV. k(x)=x+52x2+x−1k(x) = \frac{x + 5}{2x^2 + x - 1}

    • Result: No horizontal asymptote

  • Options:

    • A) I only

    • B) II only

    • C) III only

    • D) I and II only

    • E) III and IV only