Rational Functions

Definition

A rational function is any function that can be written as the quotient of two polynomial functions. Formally, if p(x) and q(x) are polynomials and q(x) \neq 0, then:

• f(x) = \frac{p(x)}{q(x)}

• The denominator q(x) must not be zero for any value in the domain.

Finding the Domain of Rational Functions

Domain Exclusion Principle

The domain of a rational function excludes all real values of x that make the denominator zero.

• For f(x) = \frac{x^2 - 25}{x - 5}, exclude x = 5 since x - 5 = 0 at x = 5.

• Domain in interval notation: (-\infty, 5) \cup (5, \infty)

• Domain in set-builder notation: \{x \mid x \neq 5\}

• For g(x) = \frac{x}{x^2 - 25}, exclude x = 5 and x = -5 since x^2 - 25 = 0 at these values.

• Domain in interval notation: (-\infty, -5) \cup (-5, 5) \cup (5, \infty)

• Domain in set-builder notation: \{x \mid x \neq 5, x \neq -5\}

• For h(x) = \frac{x+5}{x^2 + 25}, the denominator never equals zero for real x, so the domain is all real numbers.

• Domain in interval notation: (-\infty, \infty)

• Domain in set-builder notation: \{x \mid x \in \mathbb{R}\}

Graphing Rational Functions

Basic Rational Functions

The most basic rational functions are f(x) = \frac{1}{x} and f(x) = \frac{1}{x^2}. Their graphs illustrate key features such as asymptotes and end behavior.

• f(x) = \frac{1}{x} has a domain of \{x \mid x \neq 0\} and vertical/horizontal asymptotes at x = 0 and y = 0 respectively.

• Sample points: f(1) = 1, f(-1) = -1, etc.

• f(x) = \frac{1}{x^2} has a domain of \{x \mid x \neq 0\} and vertical/horizontal asymptotes at x = 0 and y = 0 respectively.

• For all real x \neq 0, f(x) > 0.

End Behavior

End behavior describes how f(x) behaves as x approaches infinity or zero:

• For f(x) = \frac{1}{x}:

  • As x \to -\infty, f(x) \to 0

  • As x \to \infty, f(x) \to 0

  • As x \to 0^-, f(x) \to -\infty

  • As x \to 0^+, f(x) \to \infty

• For f(x) = \frac{1}{x^2}:

  • As x \to 0^-, f(x) \to \infty

  • As x \to 0^+, f(x) \to \infty

  • As x \to \pm\infty, f(x) \to 0

Vertical Asymptotes

Definition and Identification

A vertical asymptote occurs at x = a if f(x) \to \infty or f(x) \to -\infty as x \to a from either side. To locate vertical asymptotes:

• Simplify f(x) if p(x) and q(x) have common factors.

• Find zeros of q(x) (after simplification); each zero gives a vertical asymptote.

Example: For f(x) = \frac{x}{x^2 - 1}, factor denominator: x^2 - 1 = (x - 1)(x + 1). Vertical asymptotes at x = 1 and x = -1.

Example: For g(x) = \frac{x - 1}{x^2 - 1}, after simplification, vertical asymptote at x = -1 and a hole at x = 1.

Example: For h(x) = \frac{x - 1}{x^2 + 1}, no vertical asymptotes since denominator never equals zero for real x.

Horizontal and Slant Asymptotes

Horizontal Asymptotes

A horizontal asymptote is a line y = b where f(x) \to b as x \to \pm\infty. The rules depend on the degrees of p(x) and q(x):

• If degree of numerator n < m (denominator), y = 0 is the horizontal asymptote.

• If n = m, y = \frac{a_n}{b_m} where a_n and b_m are leading coefficients.

• If n > m, no horizontal asymptote.

Slant (Oblique) Asymptotes

If the degree of the numerator is exactly one more than the denominator (n = m + 1), there is a slant asymptote given by the quotient of polynomial division.

• For f(x) = \frac{9x^3}{3x^2 + 1}, division yields y = 3x as the slant asymptote.

• For f(x) = \frac{x^2 + 1}{x - 1}, division yields y = x + 1 as the slant asymptote.

Using Transformations to Graph Rational Functions

Transformations

Rational functions can be graphed using transformations of basic functions:

• Shifting f(x) = \frac{1}{x} left/right or up/down by adjusting the argument and adding/subtracting constants.

• For example, g(x) = \frac{1}{x+2} - 1 is \frac{1}{x} shifted 2 units left and 1 unit down.

• h(x) = \frac{1}{x^2 + 4} is \frac{1}{x^2} shifted 4 units up.

Graphing Rational Functions: Step-by-Step

Procedure

To graph f(x) = \frac{p(x)}{q(x)}:

1. Check for symmetry: If f(-x) = f(x), the function is even and symmetric about the y-axis.

2. Find the y-intercept: Evaluate f(0).

3. Find x-intercepts: Solve p(x) = 0.

4. Find vertical asymptotes: Solve q(x) = 0.

5. Find horizontal/slant asymptotes as described above.

6. Plot additional points between and beyond intercepts and asymptotes.

7. Draw the graph, respecting asymptotes and symmetry.

Example: For f(x) = \frac{3x^2}{x^2 - 4}:

• Symmetry: f(-x) = f(x) (even function).

• Y-intercept: f(0) = 0.

• X-intercept: x = 0.

• Vertical asymptotes: x = 2 and x = -2.

• Horizontal asymptote: y = 3 (degrees equal, leading coefficients 3/1).

• Additional points: f(-3) = \frac{27}{5}, f(-1) = -1, f(1) = -1, f(3) = \frac{27}{5}.

Summary Table: Asymptote Types

Additional info: Images included are directly relevant: image_1 and image_4 show polynomial division for slant asymptotes; image_2 and image_3 show basic rational function graphs.