Rational Functions

Introduction

A rational function is a function that can be written as a quotient of two polynomials, $f (x) = \frac{P (x)}{Q (x)}$ , where $Q (x)$ is not the zero polynomial. The real domain contains exactly those real inputs for which $Q (x) \neq 0$ . Rational functions bring together algebra, graphs, limits, and modeling, so they are an important bridge from upper-secondary algebra to precalculus and calculus.

This aiMOOC is designed for Grades 11–13. You will learn to determine domains, factor and simplify expressions without losing domain restrictions, identify holes and asymptotes, analyze end behavior, sketch graphs, solve rational equations and inequalities, and use rational models in context. Extension sections connect the topic to limits, derivatives, and partial fractions.

The reciprocal function $f (x) = 1 / x$ is the basic model for many rational graphs. Its two branches illustrate inverse variation, a vertical asymptote at $x = 0$ , and a horizontal asymptote at $y = 0$ .

Learning Goals

By the end of the course, you should be able to explain what makes a function rational, state its domain, distinguish removable from non-removable discontinuities, determine intercepts and asymptotes, use degree and polynomial division to predict end behavior, and build a reliable graph from algebraic information. You should also be able to solve rational equations while rejecting invalid solutions, solve rational inequalities with sign analysis, and interpret rational functions as mathematical models.

Foundations

Definition and Domain

A rational function has the form $f (x) = P (x) / Q (x)$ . The requirement $Q (x) \neq 0$ is fundamental because division by zero is undefined. For example, $f (x) = \frac{x + 3}{x^{2} - 9}$ has denominator $(x - 3) (x + 3)$ , so the original domain excludes $x = 3$ and $x = - 3$ .

Even if a common factor cancels algebraically, the original domain restriction remains. This point distinguishes an algebraically simplified expression from the original function.

A useful habit is to record the domain before cancellation.

The Reciprocal Family

The transformed reciprocal family $f (x) = \frac{a}{x - h} + k$ is especially useful. The parameter $h$ shifts the graph horizontally, $k$ shifts it vertically, and $a$ controls vertical stretch, compression, and reflection. The vertical asymptote is $x = h$ , while the horizontal asymptote is $y = k$ .

If $a > 0$ , the branches have the same orientation as $1 / x$ . If $a < 0$ , the graph is reflected across a horizontal or vertical reference direction relative to the translated center $(h, k)$ .

Discontinuities

Holes and Removable Discontinuities

Consider $g (x) = \frac{x^{2} - 1}{x - 1}$ . Factoring gives $g (x) = \frac{(x - 1) (x + 1)}{x - 1}$ . For $x \neq 1$ , this simplifies to $g (x) = x + 1$ , but the original function is still undefined at $x = 1$ . Its graph is therefore the line $y = x + 1$ with a hole at $(1, 2)$ . This is a removable discontinuity because a single missing value would make the graph continuous there.

A common error is to cancel the factor and then forget the excluded input. To avoid this, write the restriction beside the simplified expression: $g (x) = x + 1, x \neq 1$ .

Vertical Asymptotes

A zero of the denominator that remains after all common factors are canceled usually gives a vertical asymptote. For example, $f (x) = \frac{x + 1}{x - 2}$ has a vertical asymptote at $x = 2$ because the denominator approaches zero while the numerator approaches $3$ .

Near a vertical asymptote, one-sided behavior matters. The values may approach $+ \infty$ on one side and $- \infty$ on the other, or the same infinity on both sides. A sign analysis of the factors reveals the direction.

End Behavior and Non-Vertical Asymptotes

Horizontal Asymptotes from Degrees

Let $n$ be the degree of the numerator and $m$ the degree of the denominator after you have identified any canceled factors and domain restrictions.

If $n < m$ , then the denominator grows faster and the horizontal asymptote is $y = 0$ .

If $n = m$ , the horizontal asymptote is the ratio of the leading coefficients.

If $n > m$ , there is no horizontal asymptote. Polynomial division reveals the more general end behavior.

A horizontal asymptote describes long-run behavior. A rational graph may cross a horizontal asymptote at finite values of $x$ . In contrast, the graph cannot contain a point on a vertical asymptote because that input is outside the domain.

Slant and Polynomial Asymptotes

When the numerator degree is exactly one greater than the denominator degree, polynomial division produces a linear quotient. That line is a slant or oblique asymptote.

For example, $h (x) = \frac{x^{2} + 1}{x - 1}$ can be written as $h (x) = x + 1 + \frac{2}{x - 1}$ . Because $2 / (x - 1) \to 0$ as $x \to \pm \infty$ , the graph approaches $y = x + 1$ .

If the numerator degree exceeds the denominator degree by more than one, the quotient from polynomial division is a higher-degree polynomial. The difference between the rational function and that polynomial tends to zero as $| x |$ becomes large.

Intercepts, Sign, and Local Behavior

Finding Intercepts

An x-intercept occurs where the function value is zero. For a reduced rational function, this happens at real zeros of the numerator that lie in the domain. A canceled zero does not become an x-intercept; it produces a hole.

A y-intercept exists if $x = 0$ is in the domain. Substitute $x = 0$ to find it.

For $f (x) = \frac{x + 1}{x - 2}$ , the x-intercept is $(- 1, 0)$ , and the y-intercept is $(0, - 1 / 2)$ .

Multiplicity and Sign Changes

Factored form gives more than locations. Near a zero or a vertical asymptote, multiplicity helps predict whether the sign changes.

For a factor raised to an odd power, the sign changes when crossing that critical value. For a factor raised to an even power, the sign does not change. This principle is useful for graphing and for solving rational inequalities.

A Reliable Graphing Strategy

For a rational function, use the following workflow.

Domain: Factor the denominator and exclude all values that make the original denominator zero.
Factorization: Factor numerator and denominator completely and identify common factors.
Discontinuities: Mark holes from canceled factors and vertical asymptotes from denominator factors that remain.
Intercepts: Find valid x-intercepts and the y-intercept when it exists.
End behavior: Compare degrees or use polynomial division to find horizontal, slant, or polynomial asymptotic behavior.
Sign analysis: Test intervals separated by zeros, holes, and vertical asymptotes.
Sketch: Combine all information and check that branches agree with local and end behavior.

Worked Examples

Example 1: Vertical and Horizontal Asymptotes

Analyze $f (x) = \frac{2 x + 1}{x - 3}$ .

The domain excludes $x = 3$ . There are no common factors, so $x = 3$ is a vertical asymptote. The degrees are equal, so the horizontal asymptote is the ratio of leading coefficients, $y = 2$ . The x-intercept satisfies $2 x + 1 = 0$ , giving $x = - 1 / 2$ . The y-intercept is $f (0) = - 1 / 3$ .

A useful rewrite is $f (x) = 2 + \frac{7}{x - 3}$ . This form immediately shows the translated reciprocal structure with center $(3, 2)$ .

Example 2: A Hole and a Vertical Asymptote

Analyze $g (x) = \frac{(x - 2) (x + 3)}{(x - 2) (x + 1)}$ .

The original domain excludes $x = 2$ and $x = - 1$ . Canceling the common factor gives $g (x) = \frac{x + 3}{x + 1}$ for allowed inputs. The canceled factor produces a hole at $x = 2$ . Substituting $x = 2$ into the simplified expression gives the missing y-value $5 / 3$ , so the hole is $(2, 5 / 3)$ . The uncanceled denominator factor gives a vertical asymptote at $x = - 1$ . The horizontal asymptote is $y = 1$ .

Example 3: Slant Asymptote

Analyze $p (x) = \frac{x^{2} + 3 x + 5}{x + 1}$ .

Polynomial division gives $p (x) = x + 2 + \frac{3}{x + 1}$ . Therefore the vertical asymptote is $x = - 1$ , and the slant asymptote is $y = x + 2$ . Since the numerator $x^{2} + 3 x + 5$ has discriminant $9 - 20 = - 11$ , it has no real zeros, so the graph has no x-intercepts.

Example 4: Range by Solving for Inputs

For $f (x) = \frac{x + 1}{x - 2}$ , set $y = (x + 1) / (x - 2)$ and solve for $x$ : $y x - 2 y = x + 1$ , so $x (y - 1) = 2 y + 1$ . If $y \neq 1$ , then $x = \frac{2 y + 1}{y - 1}$ . There is no solution when $y = 1$ , so the range is all real numbers except $1$ . This agrees with the transformed form $1 + 3 / (x - 2)$ . For more complicated rational functions, finding the range can require solving a polynomial equation in $x$ and checking when real solutions exist.

Rational Equations

A rational equation contains one or more rational expressions. The safest procedure is to state domain restrictions first, multiply by a least common denominator, solve the resulting equation, and then test each candidate in the original equation.

For example, solve $\frac{1}{x - 1} + \frac{1}{x + 1} = 1$ . The restrictions are $x \neq 1$ and $x \neq - 1$ . Multiplying by $(x - 1) (x + 1)$ gives $(x + 1) + (x - 1) = x^{2} - 1$ , so $2 x = x^{2} - 1$ , or $x^{2} - 2 x - 1 = 0$ . The solutions are $x = 1 \pm \sqrt{2}$ , and both satisfy the original restrictions.

Multiplying by a variable expression can create candidates that are not valid in the original equation, so the final check is essential.

Rational Inequalities

For a rational inequality such as $\frac{x - 2}{x + 1} \geq 0$ , find all critical values from numerator zeros and denominator zeros. Here they are $x = 2$ and $x = - 1$ . These values split the number line into intervals. Test one input in each interval or reason from factor signs.

The expression is positive on $(- \infty, - 1)$ , negative on $(- 1, 2)$ , and positive on $(2, \infty)$ . Because the inequality allows equality, $x = 2$ is included. Because $x = - 1$ is outside the domain, it is never included. The solution is therefore $(- \infty, - 1) \cup [2, \infty)$ .

A sign chart is usually more reliable than trying to manipulate a rational inequality by multiplying both sides by an expression whose sign is unknown.

Modeling with Rational Functions

Rational functions appear naturally when one quantity is divided by another quantity that depends on a variable.

Inverse variation has the form $y = k / x$ . If a fixed amount of work is divided among more identical workers under an idealized constant-rate assumption, the time per worker contribution can behave inversely with the number of workers. Real situations may violate this simple model, so assumptions must be stated.

Average cost is often modeled by $A (x) = C (x) / x$ , where $C (x)$ is total cost and $x > 0$ is the number of units. A fixed cost term becomes a reciprocal contribution, so average cost can decrease toward a long-run level.

Parallel resistance for two resistors satisfies $R = \frac{R_{1} R_{2}}{R_{1} + R_{2}}$ . If one resistance varies, the equivalent resistance becomes a rational function of that variable.

The thin-lens equation $\frac{1}{f} = \frac{1}{d_{o}} + \frac{1}{d_{i}}$ can be rearranged to $d_{i} = \frac{f d_{o}}{d_{o} - f}$ . The vertical asymptote at $d_{o} = f$ has a physical interpretation in the idealized model.

When you build a rational model, the mathematically allowed domain may still be wider than the physically meaningful domain. Units, positivity, measurement limits, and model assumptions matter.

Connections to Calculus

Limits and Asymptotes

Limits make asymptotic statements precise. A vertical asymptote at $x = a$ occurs when at least one one-sided limit grows without bound as $x$ approaches $a$ . A horizontal asymptote $y = L$ is described by $\lim_{x \to \infty} f (x) = L$ or $\lim_{x \to - \infty} f (x) = L$ .

For a polynomial asymptote $y = q (x)$ , the key statement is $f (x) - q (x) \to 0$ as $| x | \to \infty$ .

Derivatives and Shape

If $f (x) = P (x) / Q (x)$ and $Q (x) \neq 0$ , the quotient rule gives $f^{'} (x) = \frac{P^{'} (x) Q (x) - P (x) Q^{'} (x)}{Q (x)^{2}}$ . Critical points of the derivative help locate local maxima and minima on each domain interval. The derivative does not repair holes or vertical asymptotes; domain restrictions remain fundamental.

Partial Fractions

A proper rational function can often be decomposed into simpler rational terms. For example, $\frac{3 x + 5}{(x + 1) (x + 2)} = \frac{2}{x + 1} + \frac{1}{x + 2}$ . Such partial fraction decomposition is useful in integration, differential equations, and transform methods.

Common Misconceptions

Canceling a factor does not restore an excluded input. The original denominator determines the original domain.

A denominator zero is not automatically a vertical asymptote. A fully canceled factor can create a hole instead.

A graph may cross a horizontal or slant asymptote. These asymptotes describe end behavior, not a barrier at finite x-values.

A zero of the numerator is not always an x-intercept. The point must also belong to the domain.

Clearing denominators does not guarantee every algebraic solution is valid. Candidates must be checked against the original restrictions.

The range is not found by simply excluding every horizontal asymptote value. Some rational functions can cross and attain a horizontal asymptote; range analysis must be based on actual output values.

Interactive Tasks

Quiz: Test Your Knowledge

What is excluded from the domain of a rational function? (Inputs that make the denominator zero) (!Inputs that make the numerator zero) (!All negative inputs) (!All fractional inputs)

What does a canceled denominator factor usually produce on the graph? (A hole) (!A y-intercept) (!A horizontal asymptote) (!A turning point)

When numerator and denominator have the same degree, how is the horizontal asymptote found? (Use the ratio of leading coefficients) (!Set the numerator equal to zero) (!Use the sum of leading coefficients) (!Use the product of the degrees)

What type of asymptote occurs when the numerator degree is exactly one greater than the denominator degree? (A slant asymptote) (!A vertical asymptote) (!No asymptote of any kind) (!A circular asymptote)

Which feature can a rational graph cross? (A horizontal asymptote) (!A vertical asymptote) (!An excluded input) (!A hole)

For the function one over x minus three, where is the vertical asymptote? (x equals three) (!x equals minus three) (!y equals three) (!y equals zero)

What should you do before canceling common factors? (Record domain restrictions) (!Differentiate the function) (!Find the range) (!Complete the square)

Why are candidate solutions checked after clearing denominators? (Some candidates may be outside the original domain) (!Every rational equation has two answers) (!The numerator must stay positive) (!The graph must be symmetric)

What method is especially useful for solving rational inequalities? (A sign chart) (!A compass construction) (!A frequency table) (!A truth table)

What does polynomial division reveal for an improper rational function? (Long-run polynomial behavior) (!Only the y-intercept) (!Only the domain endpoints) (!The exact number of roots)

Memory Game

Rational function	Quotient of two polynomials with a nonzero denominator polynomial
Domain	Set of permitted input values
Hole	Missing graph point caused by a removable discontinuity
Vertical asymptote	Line approached near an excluded input with unbounded function values
Horizontal asymptote	Constant long-run output level approached at large input magnitude
Slant asymptote	Linear end-behavior line found by polynomial division
Intercept	Point where a graph meets a coordinate axis
Sign chart	Interval method for tracking positive and negative values

Drag and Drop

Match the correct terms.	Topic
Excluded denominator zero	Domain restriction
Canceled common factor	Removable discontinuity
Uncanceled denominator factor	Vertical asymptote
Equal polynomial degrees	Leading coefficient ratio
Numerator degree one higher	Slant asymptote

...

Crossword Puzzle

Domain	What set contains all permitted inputs of a function?
Asymptote	What line or curve describes limiting graph behavior?
Reciprocal	What function family contains one divided by x?
Polynomial	What type of expression appears in both parts of a rational function?
Discontinuity	What general term describes a break in a function?
Intercept	What point occurs where a graph meets an axis?

LearningApps

Cloze Text

A rational function is a quotient of two

. Its domain excludes values that make the

zero. A canceled factor can create a

. An uncanceled denominator zero can produce a vertical

. When numerator and denominator have equal degree, the horizontal asymptote uses the ratio of the leading

. When the numerator degree is exactly one greater, polynomial division gives a

asymptote. An x-intercept must come from a valid zero of the

. Rational equations require candidates to be checked against the original

. Rational inequalities are often solved with a

. In calculus, asymptotic behavior can be expressed precisely using

.

Open-Ended Tasks

Easy

Reciprocal function: Plot $y = 1 / x$ by hand, label both asymptotes, and explain in three sentences what happens as x approaches zero and as the magnitude of x becomes large.
Domain of a function: Create six rational expressions of your own and state the real domain of each, showing how you found every excluded input.
Graph transformation: Make a one-page visual showing how changing a, h, and k transforms $a / (x - h) + k$ .
Mathematical vocabulary: Record a two-minute audio or video explanation of the terms domain, hole, vertical asymptote, and horizontal asymptote using your own examples.

Standard

Graphing rational functions: Choose a rational function with at least one vertical asymptote and one x-intercept, produce a complete hand-drawn graph, and annotate every algebraic step used.
Removable discontinuity: Design two functions that simplify to the same formula but have different domains, then compare their graphs and explain why one or more holes appear.
Rational equation: Write a real-world problem that leads to a rational equation, solve it, and justify why each solution is or is not meaningful in context.
Mathematical modeling: Collect or invent a small data set that appears to follow inverse variation, fit a model of the form $y = k / x$ , and discuss where the model succeeds or fails.

Advanced

Asymptote: Build a rational function with a prescribed vertical asymptote, a prescribed hole, and a prescribed horizontal asymptote, then prove that your construction has all three features.
Polynomial division: Create three improper rational functions with linear, quadratic, and cubic polynomial asymptotes, use division to find each asymptote, and compare their end behavior graphically.
Calculus: Investigate one rational function using limits and derivatives, identify discontinuities and extrema on each domain interval, and present a complete behavior chart.
Mathematical investigation: Produce a short report or video testing the claim that a rational graph can cross a horizontal asymptote but cannot cross a vertical asymptote; include algebraic examples, graphs, and a precise conclusion.

Learning Assessment

Function analysis: Given a previously unseen rational function, determine its domain, holes, intercepts, vertical asymptotes, and end behavior, then explain how each algebraic feature appears on the graph.
Error analysis: Analyze a worked solution in which a common factor was canceled and the original domain restriction was lost; identify the error, repair the solution, and describe the graphical consequence.
Model comparison: Compare a linear model and a rational model for the same contextual data, use residual or qualitative evidence to justify which model is more appropriate, and discuss domain limitations.
Rational inequality: Solve a rational inequality using a sign chart and explain why excluded denominator zeros and included numerator zeros are treated differently.
Asymptotic reasoning: Use polynomial division and limit language to justify the end behavior of an improper rational function rather than relying only on a graphing tool.
Transfer problem: Rearrange a formula from physics, finance, or another subject into a rational function of one variable and interpret at least one domain restriction or asymptote in the original context.

Evidence of Learning

Strong evidence of learning includes accurate use of the terms rational function, domain, discontinuity, hole, intercept, and asymptote; correct factorization and simplification while preserving restrictions; graphs that agree with algebraic and asymptotic information; reliable solutions of rational equations and inequalities; and explanations that connect symbolic, graphical, numerical, and contextual representations.

Useful products include annotated graphs, sign charts, short proofs, modeling reports, videos, spreadsheets, or dynamic graphing files. At an advanced level, evidence should also include transfer: you can recognize a rational structure in a new formula, choose an efficient method, justify assumptions, and explain how algebraic restrictions affect the real situation being modeled.

OERs on the Topic

Open educational resources for further study include OpenStax Precalculus 2e: Rational Functions and OpenStax Calculus Volume 1: Limits at Infinity and Asymptotes.

Linked Learning Areas

Rational functions connect polynomial algebra with functions, graphing, equations, inequalities, limits, and mathematical modeling. Their study strengthens your ability to move between formulas and graphs, to reason about undefined values and limiting behavior, and to apply algebraic structures in calculus and science.

Rational Functions