Rational Functions

📊 What Is a Rational Function?

Part 1 of 7 — Definition, Domain & Excluded Values

A rational function is a ratio of two polynomials:

$\boxed{f(x) = \frac{p(x)}{q(x)}, \quad q(x) \neq 0}$

Just as you can't divide numbers by zero, you can't divide polynomials by zero. This makes the domain — the set of all legal inputs — the first thing to determine when working with any rational function.

📖 Recognizing Rational Functions

💡 Every polynomial is also a rational function — just one with denominator $1$.

🔑 Finding the Domain

The domain of $f(x) = \frac{p(x)}{q(x)}$ is all real numbers except where $q(x) = 0$.

Step-by-Step Process

Worked Examples

Example 1: $g(x) = \frac{3x}{x + 5}$

Set $x + 5 = 0 \implies x = -5$

$\text{Domain: } (-\infty, -5) \cup (-5, \infty)$

Example 2: $h(x) = \frac{x^2}{x^2 + 1}$

Set $x^2 + 1 = 0 \implies x^2 = -1$ — no real solutions!

$\text{Domain: } (-\infty, \infty) = \text{all real numbers}$

⚠️ Not every rational function has excluded values. If the denominator has no real roots, the domain is all reals.

Example 3: $k(x) = \frac{1}{x^3 - x}$

Factor: $x^3 - x = x(x^2 - 1) = x(x-1)(x+1)$

Excluded: $x = 0, \; x = 1, \; x = -1$

$\text{Domain: } (-\infty, -1) \cup (-1, 0) \cup (0, 1) \cup (1, \infty)$

Domain & Excluded Values Quiz 🎯

Domain Drill 🧮

1) For $f(x) = \frac{7}{x - 6}$, what value of $x$ is excluded from the domain? (e.g., for $\frac{1}{x-3}$, set $x - 3 = 0$ to get $x = 3$)

2) For $g(x) = \frac{x}{2x + 10}$, what value of $x$ is excluded? (e.g., for $\frac{x}{3x+12}$, set $3x + 12 = 0$ to get $x = -4$)

3) How many values are excluded from the domain of $h(x) = \frac{1}{x^2 + 5}$? (e.g., for $\frac{1}{x^2+1}$, $x^2 + 1 = 0$ has no real solutions → $0$ excluded)

Domain Concepts — Fill in the Blanks 🔽

Exit Quiz — Domain & Excluded Values ✅

📈 Vertical & Horizontal Asymptotes

Part 2 of 7 — Predicting Long-Run and Singular Behavior

Asymptotes are invisible boundary lines that a rational function's graph approaches but (usually) never reaches. They tell us what happens at the extremes — near excluded values and as $x \to \pm\infty$.

📖 Vertical Asymptotes

A vertical asymptote occurs at $x = c$ when:

1. $q(c) = 0$ (denominator is zero), AND
2. The factor $(x - c)$ does not cancel with the numerator

$\boxed{\text{VA at } x = c \iff q(c) = 0 \text{ and } p(c) \neq 0}$

What Happens Near a VA

As $x$ approaches $c$, $f(x) \to +\infty$ or $f(x) \to -\infty$ (the graph shoots up or down).

Worked Example

Find the vertical asymptote(s) of $f(x) = \frac{2x}{x^2 - 1}$.

Step 1: Factor denominator: $x^2 - 1 = (x-1)(x+1)$

Step 2: Set each factor to zero: $x = 1$ and $x = -1$

Step 3: Check numerator: $2(1) = 2 \neq 0$ and $2(-1) = -2 \neq 0$

Result: Vertical asymptotes at $x = 1$ and $x = -1$

⚠️ If both numerator and denominator are zero at $x = c$, the common factor cancels and you get a hole (Part 3), not a vertical asymptote.

📖 Horizontal Asymptotes

A horizontal asymptote tells you the output value that $f(x)$ approaches as $x \to \pm\infty$. It depends entirely on comparing the degrees of the numerator and denominator.

Worked Examples

Example 1: $f(x) = \frac{3x + 1}{x^2 + 5}$ → $\deg(p) = 1 < \deg(q) = 2$ → HA: $y = 0$

Example 2: $g(x) = \frac{4x^2 - 1}{2x^2 + 3}$ → $\deg(p) = \deg(q) = 2$ → HA: $y = \frac{4}{2} = 2$

Example 3: $h(x) = \frac{x^3}{x + 1}$ → $\deg(p) = 3 > \deg(q) = 1$ → No HA (slant asymptote exists)

💡 Memory aid: "Bottom wins → $y = 0$. Tie → ratio of leaders. Top wins → no HA."

📐 Slant (Oblique) Asymptotes

When $\deg(p) = \deg(q) + 1$ (numerator is exactly one degree higher), the function has a slant asymptote found by polynomial long division.

Example

Find the slant asymptote of $f(x) = \frac{x^2 + 3x + 5}{x + 1}$.

Divide $x^2 + 3x + 5$ by $x + 1$:

$x^2 + 3x + 5 = (x + 1)(x + 2) + 3$

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

As $x \to \pm\infty$, $\frac{3}{x+1} \to 0$, so:

$\boxed{\text{Slant asymptote: } y = x + 2}$

Asymptote Quiz 🎯

Asymptote Drill 🧮

1) What is the horizontal asymptote of $f(x) = \frac{6x^2 + 1}{3x^2 - 7}$? Give the $y$-value. (e.g., for $\frac{8x^2}{4x^2 + 1}$, HA is $y = \frac{8}{4} = 2$)

2) How many vertical asymptotes does $g(x) = \frac{1}{x^2 - 5x + 6}$ have? (e.g., for $\frac{1}{x^2-1}$, factor to $(x-1)(x+1)$ → $2$ VAs)

3) For $h(x) = \frac{x^2 + x}{x + 1}$, after canceling the common factor, what is $h(x)$ simplified? Give just the simplified expression as a number (evaluate $h(2)$). (e.g., $\frac{x(x+3)}{x+3} = x$, so $h(2) = 2$)

Asymptote Rules — Fill in the Blanks 🔽

Exit Quiz — Asymptotes ✅

🕳️ Holes & Removable Discontinuities

Part 3 of 7 — When Factors Cancel

Not every denominator zero produces a vertical asymptote. When numerator and denominator share a common factor, canceling it creates a hole — a single missing point where the function is undefined but the graph has no dramatic blow-up.

📖 Holes vs. Vertical Asymptotes

When $q(c) = 0$, there are two possibilities:

The Key Principle

$\boxed{\frac{(x - c) \cdot g(x)}{(x - c) \cdot h(x)} = \frac{g(x)}{h(x)}, \quad x \neq c}$

Canceling removes the factor from the expression, but the restriction $x \neq c$ remains. The function is still undefined at $x = c$.

🔑 Finding Hole Coordinates

A hole is a point, not just an $x$-value. To find the full coordinates:

Worked Example with Multiple Features

Analyze $f(x) = \frac{x^2 - x - 6}{x^2 - 9}$ completely.

Factor: $\frac{(x-3)(x+2)}{(x-3)(x+3)}$

Common factor: $(x - 3)$ → hole at $x = 3$

After canceling: $f(x) = \frac{x+2}{x+3}, \quad x \neq 3$

Hole coordinates: $f(3) = \frac{3+2}{3+3} = \frac{5}{6}$ → Hole at $\left(3, \frac{5}{6}\right)$

Remaining denominator: $x + 3 = 0$ → VA at $x = -3$

HA: $\deg(p) = \deg(q) = 1$ → $y = \frac{1}{1} = 1$

💡 One function can have BOTH holes and vertical asymptotes — they come from different factors.

Holes Quiz 🎯

Hole Coordinates Drill 🧮

1) For $f(x) = \frac{x^2 - 9}{x + 3}$, what is the $y$-coordinate of the hole? (e.g., $\frac{x^2-4}{x-2} = x + 2$ for $x \neq 2$, so the hole $y$-value is $2 + 2 = 4$)

2) How many holes does $g(x) = \frac{(x-1)(x+2)}{(x-1)(x+2)(x-3)}$ have? (e.g., $\frac{(x-1)}{(x-1)(x+4)}$ has $1$ hole at $x = 1$)

3) For $h(x) = \frac{x^2 - 5x + 6}{x - 2}$, what is the $x$-intercept of the simplified function? (e.g., $\frac{(x-1)(x-3)}{x-1} = x - 3$, so $x$-intercept is $3$)

Holes vs. Asymptotes — Fill in the Blanks 🔽

Exit Quiz — Holes & Removable Discontinuities ✅

✂️ Simplifying Rational Expressions

Part 4 of 7 — Algebraic Simplification, Addition, & Division

Before you can graph or analyze a rational function, you often need to simplify it first. This part covers the algebraic mechanics: factoring and canceling, adding/subtracting with common denominators, multiplying/dividing rational expressions, and rewriting via polynomial long division.

📖 Factoring & Canceling

The fundamental simplification technique:

$\boxed{\frac{A \cdot C}{B \cdot C} = \frac{A}{B}, \quad C \neq 0}$

Step-by-Step

⚠️ Never cancel terms — only factors! $\frac{x + 3}{x + 5} \neq \frac{3}{5}$. You can only cancel something that multiplies the entire numerator and entire denominator.

🔧 Operations with Rational Expressions

Adding & Subtracting (LCD Method)

$\frac{A}{B} + \frac{C}{D} = \frac{AD + BC}{BD}$

Example: $\frac{2}{x-1} + \frac{3}{x+4}$

LCD $= (x-1)(x+4)$

$= \frac{2(x+4) + 3(x-1)}{(x-1)(x+4)} = \frac{2x + 8 + 3x - 3}{(x-1)(x+4)} = \frac{5x + 5}{(x-1)(x+4)} = \frac{5(x+1)}{(x-1)(x+4)}$

Multiplying & Dividing

💡 Always factor before multiplying — it makes cancellation much easier.

✏️ Polynomial Long Division for Rationals

When $\deg(p) \geq \deg(q)$, you can rewrite $\frac{p(x)}{q(x)}$ as:

$\frac{p(x)}{q(x)} = \text{quotient} + \frac{\text{remainder}}{q(x)}$

This is essential for finding slant asymptotes and understanding end behavior.

Worked Example

Rewrite $\frac{2x^2 + 3x - 5}{x + 2}$ in quotient-remainder form.

Dividing:

• $2x^2 \div x = 2x$. Multiply: $2x(x+2) = 2x^2 + 4x$. Subtract: $(2x^2 + 3x) - (2x^2 + 4x) = -x$.
• Bring down: $-x - 5$. Divide: $-x \div x = -1$. Multiply: $-1(x+2) = -x - 2$. Subtract: $(-x-5) - (-x-2) = -3$.

$\boxed{\frac{2x^2 + 3x - 5}{x + 2} = 2x - 1 + \frac{-3}{x + 2}}$

As $x \to \pm\infty$, $\frac{-3}{x+2} \to 0$, so the slant asymptote is $y = 2x - 1$.

Simplification Quiz 🎯

Simplification Drill 🧮

1) Simplify $\frac{x^2 - 25}{x + 5}$ and evaluate at $x = 3$. (e.g., $\frac{x^2-4}{x+2} = x - 2$, so at $x = 3$: $3 - 2 = 1$)

2) What is $\frac{2}{x} + \frac{3}{x}$? Evaluate at $x = 5$. (e.g., $\frac{1}{x} + \frac{4}{x} = \frac{5}{x}$, so at $x = 5$: $\frac{5}{5} = 1$)

3) Divide: $\frac{x^2 + 2x + 1}{x + 1}$. Evaluate at $x = 4$. (e.g., $\frac{x^2+6x+9}{x+3} = x + 3$, so at $x = 4$: $7$)

Simplification Rules — Fill in the Blanks 🔽

Exit Quiz — Simplification ✅

📉 Graphing Rational Functions

Part 5 of 7 — Transformations & Complete Graph Sketching

Graphing a rational function means assembling all the pieces from Parts 1–4: domain, intercepts, asymptotes, holes, and sign behavior. This part gives you a systematic graphing procedure and introduces transformations of the parent function $y = \frac{1}{x}$.

📖 The Parent Function $y = \frac{1}{x}$

Every simple rational function is a transformation of this parent graph.

Transformation Form

$\boxed{f(x) = \frac{a}{x - h} + k}$

Example: $f(x) = \frac{-2}{x + 1} + 3$ has VA at $x = -1$, HA at $y = 3$, reflected and stretched by 2.

📋 Complete Graphing Procedure

Follow these steps for any rational function $f(x) = \frac{p(x)}{q(x)}$:

Worked Example

Sketch $f(x) = \frac{x - 1}{x^2 - 4}$

1. Factor: $\frac{x-1}{(x-2)(x+2)}$ — no common factors
2. Domain: $x \neq 2, x \neq -2$
3. Holes: None
4. VAs: $x = 2$ and $x = -2$
5. HA: $\deg(p) = 1 < \deg(q) = 2$ → $y = 0$
6. $x$-intercept: $x - 1 = 0 \implies x = 1$ → point $(1, 0)$
7. $y$-intercept: $f(0) = \frac{-1}{-4} = \frac{1}{4}$ → point $\left(0, \frac{1}{4}\right)$
8. Sign analysis: Test in intervals $(-\infty, -2)$, $(-2, 1)$, $(1, 2)$, $(2, \infty)$

Graphing Quiz 🎯

Graphing Features Drill 🧮

1) What is the $y$-intercept of $f(x) = \frac{x + 3}{x - 1}$? Give the $y$-value. (e.g., for $\frac{x+2}{x-4}$, $f(0) = \frac{2}{-4} = -0.5$)

2) For $g(x) = \frac{5}{x + 4} - 2$, what is the horizontal asymptote? Give the $y$-value. (e.g., $\frac{3}{x-1} + 7$ has HA at $y = 7$)

3) How many vertical asymptotes does $h(x) = \frac{x}{x^3 - x}$ have after simplification? (e.g., $\frac{x}{x^2-x} = \frac{x}{x(x-1)} = \frac{1}{x-1}$ has $1$ VA)

Graphing Concepts — Fill in the Blanks 🔽

Exit Quiz — Graphing ✅

⚖️ Rational Equations & Inequalities

Part 6 of 7 — Solving Rational Equations, Checking for Extraneous Solutions, and Rational Inequalities

Up to now we have analyzed rational functions. This part shifts to solving — finding $x$-values that satisfy rational equations and inequalities. The critical new skill is checking for extraneous solutions introduced when you multiply both sides by an expression containing the variable.

📖 Solving Rational Equations

The LCD Method

⚠️ Step 6 is mandatory. Multiplying by the LCD can introduce false solutions that make the original denominator zero.

🚨 Extraneous Solutions

What Are They?

An extraneous solution is a value that satisfies the transformed equation but makes a denominator in the original equation equal to zero.

Example: Extraneous Solution in Action

Solve $\frac{x}{x - 3} = \frac{3}{x - 3} + 2$

Step 1: LCD $= (x - 3)$. Multiply through: $x = 3 + 2(x - 3)$

Step 2: Simplify: $x = 3 + 2x - 6 = 2x - 3$ $-x = -3 \implies x = 3$

Step 3: CHECK. The original equation has $x - 3$ in the denominator.

At $x = 3$: denominator $= 3 - 3 = 0$ ❌ Undefined!

$\boxed{\text{No solution (the only candidate is extraneous)}}$

💡 Always check your answers against the original equation's domain restrictions.

📊 Rational Inequalities

For inequalities like $\frac{p(x)}{q(x)} > 0$ or $\frac{p(x)}{q(x)} \leq 0$, use a sign chart:

Example

Solve $\frac{x - 1}{x + 3} \geq 0$

Critical values: $x = 1$ (numerator = 0) and $x = -3$ (denominator = 0)

$\geq 0$: want positive or zero. Include $x = 1$ (zero), exclude $x = -3$ (undefined).

$\boxed{(-\infty, -3) \cup [1, \infty)}$

Equations & Inequalities Quiz 🎯

Solving Drill 🧮

1) Solve: $\frac{6}{x} = 2$. What is $x$? (e.g., $\frac{10}{x} = 5$ → $x = \frac{10}{5} = 2$)

2) How many extraneous solutions arise when solving $\frac{x^2 - 4}{x - 2} = x + 2$? (e.g., if the only solution makes a denominator zero, that is $1$ extraneous solution)

3) For $\frac{x+1}{x-3} > 0$, how many intervals are in the solution set? (e.g., $\frac{x}{x-1} > 0$ has solution $(-\infty, 0) \cup (1, \infty)$ — that is $2$ intervals)

Solving Rules — Fill in the Blanks 🔽

Exit Quiz — Rational Equations & Inequalities ✅

🏆 Rational Functions — Full Synthesis

Part 7 of 7 — Putting It All Together

This final part integrates every concept: domain, asymptotes, holes, simplification, graphing, and solving. The problems are multi-step — just like exam questions.

Your Rational Functions Toolkit

📋 Complete Rational Function Analysis

Worked Example

Fully analyze $f(x) = \frac{2x^2 - 2}{x^2 - 4x + 3}$.

Step 1 — Factor completely: $f(x) = \frac{2(x^2 - 1)}{(x-1)(x-3)} = \frac{2(x-1)(x+1)}{(x-1)(x-3)}$

Step 2 — Identify features:

✏️ From Graph to Equation

When given a rational function's graph, work backwards:

Example

A rational function has VA at $x = -2$, HA at $y = 3$, $x$-intercept at $x = 1$, and $y$-intercept at $(0, -\frac{3}{2})$.

Build the equation:

VA at $x = -2$ → denominator has $(x + 2)$

$x$-intercept at $x = 1$ → numerator has $(x - 1)$

HA at $y = 3$ → equal degrees, ratio of leading coefficients is $3$

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

Verify $y$-intercept: $f(0) = \frac{3(-1)}{2} = -\frac{3}{2}$ ✔

Synthesis Quiz 🎯

Multi-Step Drill 🧮

1) For $f(x) = \frac{x^2 - 9}{x + 3}$, what is the $y$-coordinate of the hole? (e.g., $\frac{x^2 - 4}{x - 2} = x + 2$ for $x \neq 2$, hole $y$-value: $2 + 2 = 4$)

2) What is the horizontal asymptote of $\frac{3x^3 + 1}{6x^3 - x}$? Give the $y$-value. (e.g., $\frac{4x^2}{2x^2+1}$ has HA $y = \frac{4}{2} = 2$)

3) How many vertical asymptotes does $\frac{x^2 - 1}{x^3 - x}$ have? (e.g., $\frac{x}{x^2 - x} = \frac{1}{x - 1}$ has $1$ VA)

Synthesis — Match the Strategy 🔽

Final Exit Quiz — Rational Functions ✅