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Rational Functions

Introduction to Rational Functions

Rational functions are functions of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. A basic example is the reciprocal function f(x) = 1/x. Understanding the behavior, graphs, and limits of rational functions is essential in College Algebra.

• Key Term: Reciprocal Function – A function of the form f(x) = 1/x.

• Key Term: Asymptote – A line that a graph approaches but never touches.

Tables and Graphs of Reciprocal Functions

To analyze rational functions, we often start by creating tables of values and sketching their graphs.

• As x becomes very large (positive or negative), f(x) = 1/x approaches 0.

• As x approaches 0 from the positive side, f(x) increases without bound (goes to infinity).

• As x approaches 0 from the negative side, f(x) decreases without bound (goes to negative infinity).

x	f(x) = 1/x
-3	-0.33
-2	-0.5
-1	-1
-0.5	-2
0.5	2
1	1
2	0.5
3	0.33

Example: The graph of f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

Arrow Notation and End Behavior

Arrow notation is used to describe the end behavior and local behavior of functions, especially near asymptotes.

Arrow Notation	Meaning
x → ∞	x approaches infinity (right)
x → -∞	x approaches negative infinity (left)
x → a⁺	x approaches a from the right
x → a⁻	x approaches a from the left
f(x) → ∞	f(x) increases without bound
f(x) → -∞	f(x) decreases without bound

Vertical and Horizontal Asymptotes

Asymptotes are lines that the graph of a function approaches but does not cross (in most cases).

• Vertical Asymptote: The line x = a is a vertical asymptote of f(x) if f(x) becomes arbitrarily large as x approaches a.

• Horizontal Asymptote: The line y = b is a horizontal asymptote of f(x) if f(x) approaches b as x approaches infinity or negative infinity.

Example: For f(x) = 1/(x-2), the vertical asymptote is x = 2, and the horizontal asymptote is y = 0.

Transformations of Rational Functions

Rational functions can be transformed by shifting, stretching, or reflecting their graphs.

• Shifting left/right: f(x) = 1/(x-h) shifts the graph h units right.

• Shifting up/down: f(x) = 1/x + k shifts the graph k units up.

• Reflection: f(x) = -1/x reflects the graph across the x-axis.

Example: f(x) = 1/(x-2) + 3 has a vertical asymptote at x = 2 and a horizontal asymptote at y = 3.

Domain and Range of Rational Functions

The domain of a rational function is all real numbers except where the denominator is zero. The range depends on the function's behavior and asymptotes.

• Domain: For f(x) = 1/(x-2), domain is all real numbers except x = 2.

• Range: For f(x) = 1/(x-2), range is all real numbers except y = 0.

Limits of Rational Functions

Two-Sided Limits

A two-sided limit describes the value that a function approaches as x approaches a specific value from both sides.

• If the function is undefined at x = a, but approaches the same value from both sides, the two-sided limit exists.

• Example: For f(x) = (x^2 - 4)/(x - 2), as x approaches 2, f(x) approaches 4.

$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$

One-Sided Limits

One-sided limits consider the behavior of a function as x approaches a value from only one side (left or right).

• Left-hand limit: $\lim_{x \to a^-} f(x)$

• Right-hand limit: $\lim_{x \to a^+} f(x)$

• If the left and right limits are not equal, the two-sided limit does not exist.

• Example: For f(x) = (x-2)/(x-2), as x approaches 2, the function is undefined, but the left and right limits can be considered separately.

Infinite Limits

Infinite limits occur when the function increases or decreases without bound as x approaches a certain value.

• Example: For f(x) = 1/(x-2)^2, as x approaches 2, f(x) approaches infinity from both sides.

$\lim_{x \to 2} \frac{1}{(x-2)^2} = \infty$

Practice Problems and Applications

Analyzing Transformations and Asymptotes

Practice involves identifying equations of asymptotes, describing transformations, and determining domain and range from graphs.

• Given f(x) = 1/(x-2) - 1, vertical asymptote: x = 2, horizontal asymptote: y = -1.

• Domain: all real numbers except x = 2.

• Range: all real numbers except y = -1.

Evaluating Limits from Graphs

Limits can be estimated by observing the behavior of the graph near the point of interest.

• For a function with a jump or infinite discontinuity, use one-sided limits.

• For removable discontinuities, the limit may exist even if the function is undefined at that point.

Summary Table: Asymptotes and Transformations

Function	Vertical Asymptote	Horizontal Asymptote	Domain	Range
f(x) = 1/x	x = 0	y = 0	All x ≠ 0	All y ≠ 0
f(x) = 1/(x-2)	x = 2	y = 0	All x ≠ 2	All y ≠ 0
f(x) = 1/(x-2) + 3	x = 2	y = 3	All x ≠ 2	All y ≠ 3
f(x) = 1/(x+1)	x = -1	y = 0	All x ≠ -1	All y ≠ 0

Key Formulas

• General rational function: $f(x) = \frac{a}{x-h} + k$

• Vertical asymptote: Set denominator = 0, solve for x.

• Horizontal asymptote: For $f(x) = \frac{a}{x-h} + k$, y = k.

Additional info:

• These notes cover rational functions, their graphs, asymptotes, and limits, which are central topics in College Algebra (Chapters 4, 5, and 9).

• Limit notation and graphical analysis are foundational for calculus, but are introduced here for rational functions.