Rational Functions and Asymptotes

📐 Domain Restrictions

Part 1 of 7 — Domain Restrictions

A rational function is $f(x) = \frac{p(x)}{q(x)}$ where $q(x) \neq 0$.

Domain: all real numbers except where the denominator = 0.

Set $q(x) = 0$ and solve to find restrictions.

Worked Example

$f(x) = \frac{x+1}{x-3}$. Domain?

$x - 3 = 0 \Rightarrow x = 3$

Domain: all reals except $x = 3$ ✅

Concept Check 🎯

Domain Restrictions 🧮

Find where denominator = 0:

1. $\frac{1}{x-5}$. Restricted at $x =$ ?

2. $\frac{x}{x+2}$. Restricted at $x =$ ?

3. $\frac{1}{x^2-4}$. One restriction at $x =$ ?

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📊 Vertical Asymptotes

Part 2 of 7 — Vertical Asymptotes

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

• The denominator equals zero at $x = a$
• The factor does NOT cancel with the numerator

The graph approaches $\pm\infty$ near a vertical asymptote.

Worked Example

$f(x) = \frac{2}{x-4}$. Vertical asymptote?

$x - 4 = 0 \Rightarrow x = 4$

Vertical asymptote at $x = 4$ ✅

Concept Check 🎯

Vertical Asymptotes 🧮

1. $\frac{1}{x-3}$. VA at $x =$ ?

2. $\frac{x}{x+1}$. VA at $x =$ ?

3. $\frac{1}{(x-2)(x+5)}$. One VA at $x =$ ?

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🔢 Horizontal Asymptotes

Part 3 of 7 — Horizontal Asymptotes

Compare degrees of numerator ($n$) and denominator ($d$):

Worked Example

$f(x) = \frac{3x^2}{x^2 + 1}$. HA?

Degrees equal (both 2). HA: $y = \frac{3}{1} = 3$ ✅

Concept Check 🎯

Horizontal Asymptotes 🧮

1. $\frac{1}{x+1}$. HA: $y =$ ?

2. $\frac{2x}{x-3}$. HA: $y =$ ?

3. $\frac{4x^2}{2x^2+1}$. HA: $y =$ ?

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📈 Holes in Graphs

Part 4 of 7 — Holes in Graphs

A hole occurs when a factor cancels from both numerator and denominator.

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

The $(x-2)$ cancels → hole at $x = 2$.

To find the y-value of the hole, substitute into the simplified function.

Worked Example

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

Cancel $(x-2)$: simplified = $\frac{x+1}{x+3}$

Hole at $x=2$: $y = \frac{2+1}{2+3} = \frac{3}{5} = 0.6$ ✅

Concept Check 🎯

Holes 🧮

1. $\frac{(x-3)(x+1)}{(x-3)(x-1)}$. Hole at $x =$ ?

2. $\frac{x(x-5)}{x(x+2)}$. Hole at $x =$ ?

3. $\frac{(x-2)(x+1)}{(x-2)(x+3)}$. y-value of hole?

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🧮 Graphing Rational Functions

Part 5 of 7 — Graphing Rational Functions

Steps:

1. Find domain restrictions (den = 0)
2. Identify holes (cancel common factors)
3. Find VAs (remaining den zeros)
4. Find HA (compare degrees)
5. Find x-intercepts (num = 0) and y-intercept ($f(0)$)
6. Plot and connect

Worked Example

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

• No common factors → no holes
• VA: $x = 1$
• HA: $y = 1$ (equal degrees, 1/1)
• x-int: $x = -2$
• y-int: $f(0) = \frac{2}{-1} = -2$ ✅

Concept Check 🎯

Graph $\frac{x}{x-2}$ 🧮

1. VA at $x =$ ?

2. HA at $y =$ ?

3. x-intercept at $x =$ ?

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🛠️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Combine all concepts for rational functions:

• Domain, VAs, HAs, holes
• Intercepts
• Sketch the graph

Worked Example

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

• VAs: $x = 1, x = -1$
• HA: $y = 0$ (degree 1 < degree 2)
• x-int: $x = 0$, y-int: $f(0) = 0$ ✅

Concept Check 🎯

Analyze $\frac{x-3}{x+2}$ 🧮

1. VA at $x =$ ?

2. HA at $y =$ ?

3. x-intercept at $x =$ ?

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🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Concepts

• Domain: exclude den = 0
• VA: non-canceled den zeros
• HA: compare degrees ($n<d$: y=0; $n=d$: LC ratio; $n>d$: none)
• Holes: canceled common factors
• x-int: num = 0; y-int: f(0)

Worked Example

$\frac{3x+6}{x+2} = \frac{3(x+2)}{x+2}$

Common factor → hole at $x=-2$, simplified: $y = 3$ (Horizontal line with a hole) ✅

Concept Check 🎯

Review 🧮

1. VA of $\frac{5}{x-4}$ at $x =$ ?

2. HA of $\frac{2x}{3x+1}$ at $y =$ ? (round to hundredths)

3. Hole of $\frac{x(x-1)}{x(x+3)}$ at $x =$ ?

Concept Check 🔍

Practice

Challenge Question 📋