Rational Functions

Domain and Vertical Asymptotes

A rational function is a ratio of two polynomials \(\frac{P(x)}{Q(x)}\). Understanding its domain and asymptotes is crucial for graphing and analysis.

Domain Analysis:

    The domain includes all real numbers except where \(Q(x) = 0\)

    To find excluded values, factor \(Q(x)\) and solve \(Q(x) = 0\)

    Each x-value that makes \(Q(x) = 0\) creates a vertical asymptote, unless it's also a root of \(P(x)\)

Example:

For \(f(x) = \frac{x^2 - 4}{x - 2}\):

    Factor numerator: \(\frac{(x+2)(x-2)}{x-2}\)

    Cancel common factor: \(x + 2\)

    Domain: \(x \neq 2\)

    Note: This creates a hole at \(x = 2\)

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of rational functions as \(x\) approaches \(\pm\infty\).

Three Cases:

    If degree of \(P(x)\) < degree of \(Q(x)\):

      Horizontal asymptote at \(y = 0\)

      Example: \(f(x) = \frac{2x}{x^2 + 1}\) has HA: \(y = 0\)

    If degree of \(P(x)\) = degree of \(Q(x)\):

      HA at \(y = \frac{\text{leading coefficient of }P(x)}{\text{leading coefficient of }Q(x)}\)

      Example: \(f(x) = \frac{3x^2 + 1}{x^2 - 2}\) has HA: \(y = 3\)

    If degree of \(P(x)\) > degree of \(Q(x)\):

      No horizontal asymptote

      Function grows without bound

Graphing Rational Functions

To graph rational functions, follow these steps:

    Find domain (excluded values)

    Find vertical asymptotes

    Find horizontal/oblique asymptotes

    Find x and y-intercepts

    Find any holes

    Plot additional points and sketch the curve

Example: \(f(x) = \frac{x^2 - 1}{x - 1}\)

Key features:

    Domain: \(x \neq 1\)

    Vertical asymptote: \(x = 1\)

    Horizontal asymptote: \(y = x\)

    Hole at \((1, 2)\)

Piecewise Rational Functions

Piecewise rational functions combine different rational expressions over different intervals.

Key concepts:

    Finding domain for each piece

    Identifying discontinuities at transition points

    Analyzing end behavior for each piece

    Analyzing behavior near transitions

Example: \[f(x) = \begin{cases} \frac{x^2}{x-1} & \text{for } x < 1 \\ \frac{x+2}{x-2} & \text{for } x > 1 \end{cases}\]

Key features:

    Discontinuity at \(x = 1\)

    Vertical asymptotes at \(x = 1\) and \(x = 2\)

    Different end behaviors for each piece

    Domain: \((-\infty,1) \cup (1,2) \cup (2,\infty)\)

Comprehensive Practice

Test your knowledge of all rational function concepts.

Practice: Rational Functions