# Domain and Range of Rational Functions - Tutorial
## Algebra 2 - Chapter 7
**Understanding what $x$-values are allowed and what $y$-values are possible**
*A Complete Tutorial with Examples*
---
## Slide 1: What is Domain?
:::info
**Definition of Domain**
The **domain** of a function is the set of all possible *input values* ($x$-values) for which the function is defined.
:::
### Think of it this way:
- Domain = All the $x$-values you can "plug in" to the function
- Domain = All the $x$-values that make the function work
- Domain = The horizontal extent of the graph
### Real-World Analogy 🎮
Think of a video game where you can move left and right. The domain is like the boundaries of the screen - it tells you how far left and right you can go!
---
## Slide 2: What is Range?
:::info
**Definition of Range**
The **range** of a function is the set of all possible *output values* ($y$-values) that the function can produce.
:::
### Think of it this way:
- Range = All the $y$-values the function can output
- Range = All the $y$-values that appear on the graph
- Range = The vertical extent of the graph
### Real-World Analogy 🎮
Continuing with the video game: the range is like how high and low you can jump. It tells you the vertical boundaries of where you can be!
---
## Slide 3: Why Rational Functions Are Special
### A rational function has the form:
$$
f(x) = \frac{p(x)}{q(x)}
$$
where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$
:::danger
**🚨 The Critical Rule**
**You can NEVER divide by zero!**
This means any $x$-value that makes the denominator equal to zero must be EXCLUDED from the domain.
:::
:::success
**💡 Key Insight:** The denominator controls the domain!
:::
---
## Slide 4: Finding Domain - Step-by-Step
### The Process:
**Step 1: Identify the denominator**
- Look at the bottom of your fraction
**Step 2: Set the denominator equal to zero**
- Solve: denominator $= 0$
**Step 3: Solve for $x$**
- Find the value(s) of $x$ that make the denominator zero
**Step 4: Exclude those $x$-values**
- State: "All real numbers except $x = $ ___"
---
## Slide 5: Example 1 - Finding Domain
:::warning
**Problem:**
$$
f(x) = \frac{1}{x}
$$
:::
### Solution:
**Step 1:** Denominator is $x$
**Step 2:** Set denominator $= 0$
$$
x = 0
$$
**Step 3:** $x = 0$ makes denominator zero
**Step 4:** Exclude $x = 0$
:::success
**✅ Answer:**
**Domain:** All real numbers except $x = 0$
Or in notation: $\{x \mid x \neq 0\}$
:::
---
## Slide 6: Example 2 - Domain with Translation
:::warning
**Problem:**
$$
g(x) = \frac{3}{x - 2}
$$
:::
### Solution:
**Step 1:** Denominator is $(x - 2)$
**Step 2:** Set denominator $= 0$
$$
x - 2 = 0
$$
**Step 3:** Solve for $x$
$$
x = 2
$$
**Step 4:** Exclude $x = 2$
:::success
**✅ Answer:**
**Domain:** All real numbers except $x = 2$
Or in notation: $\{x \mid x \neq 2\}$
:::
---
## Slide 7: Example 3 - More Complex Domain
:::warning
**Problem:**
$$
h(x) = \frac{2x + 1}{x - 3}
$$
:::
### Solution:
**Step 1:** Denominator is $(x - 3)$
**Step 2:** Set denominator $= 0$
$$
x - 3 = 0
$$
**Step 3:** Solve for $x$
$$
x = 3
$$
**Step 4:** Exclude $x = 3$
:::success
**✅ Answer:**
**Domain:** All real numbers except $x = 3$
**Notice:** The numerator doesn't affect the domain!
:::
---
## Slide 8: Asymptotes - The Key to Range
:::info
**What is an Asymptote?**
An **asymptote** is a line that the graph approaches but never touches or crosses.
:::
### Vertical Asymptote
- A vertical line: $x = a$
- Where denominator $= 0$
- Graph goes to $\pm\infty$
- **Determines DOMAIN restrictions**
### Horizontal Asymptote
- A horizontal line: $y = b$
- What $y$ approaches as $x \to \pm\infty$
- Graph gets closer but doesn't touch
- **Determines RANGE restrictions**
:::danger
**Important Connection!**
- Vertical asymptote at $x = a$ $\Rightarrow$ Domain excludes $x = a$
- Horizontal asymptote at $y = b$ $\Rightarrow$ Range excludes $y = b$
:::
---
## Slide 9: Finding Range - Step-by-Step
### The Process:
**Step 1: Identify the form of the function**
- For $y = \frac{a}{x-h} + k$, the horizontal asymptote is $y = k$
**Step 2: Find the horizontal asymptote**
- For $y = \frac{a}{x-h} + k$: horizontal asymptote is $y = k$
- For $y = \frac{ax+b}{cx+d}$: horizontal asymptote is $y = \frac{a}{c}$
**Step 3: Determine which $y$-values the graph can reach**
- The graph can reach all $y$-values EXCEPT the horizontal asymptote
**Step 4: State the range**
- "All real numbers except $y = $ ___"
---
## Slide 10: Example 4 - Finding Range
:::warning
**Problem:**
$$
f(x) = \frac{1}{x}
$$
:::
### Solution:
**Step 1:** This is the parent function form
**Step 2:** Horizontal asymptote is $y = 0$
- (As $x$ gets very large, $y$ approaches $0$)
**Step 3:** The function can output any $y$-value except $0$
**Step 4:** Range excludes $y = 0$
:::success
**✅ Answer:**
**Range:** All real numbers except $y = 0$
Or in notation: $\{y \mid y \neq 0\}$
:::
---
## Slide 11: Example 5 - Range with Translation
:::warning
**Problem:**
$$
g(x) = \frac{-4}{x + 2} - 1
$$
This is in the form: $y = \frac{a}{x-h} + k$
:::
### Solution:
**Step 1:** Identify $k$ value (vertical shift)
$$
k = -1
$$
**Step 2:** Horizontal asymptote is $y = k$
$$
y = -1
$$
**Step 3:** The graph shifted down 1 unit
**Step 4:** Range excludes $y = -1$
:::success
**✅ Answer:**
**Range:** All real numbers except $y = -1$
:::
---
## Slide 12: Example 6 - Complete Analysis
:::warning
**Problem: Find domain and range**
$$
f(x) = \frac{2x + 1}{x - 3}
$$
:::
### Finding Domain:
1. Denominator: $x - 3$
2. Set equal to zero: $x - 3 = 0$
3. Solve: $x = 3$
4. Vertical asymptote: $x = 3$
:::success
**Domain:** All reals except $x = 3$
:::
### Finding Range:
1. Form: $\frac{ax+b}{cx+d}$
2. Horizontal asymptote: $y = \frac{a}{c}$
3. Calculate: $y = \frac{2}{1} = 2$
4. Graph approaches but never reaches $y = 2$
:::success
**Range:** All reals except $y = 2$
:::
---
## Slide 13: Quick Reference Table
| Function Form | Vertical Asymptote | Horizontal Asymptote |
|---------------|-------------------|---------------------|
| $y = \frac{a}{x}$ | $x = 0$ | $y = 0$ |
| $y = \frac{a}{x-h}$ | $x = h$ | $y = 0$ |
| $y = \frac{a}{x-h} + k$ | $x = h$ | $y = k$ |
| $y = \frac{ax+b}{cx+d}$ | $x = -\frac{d}{c}$ | $y = \frac{a}{c}$ |
:::success
**Remember:**
- Domain excludes vertical asymptote values
- Range excludes horizontal asymptote values
:::
---
## Slide 14: Common Mistakes to Avoid
:::danger
**❌ Mistake 1: Confusing Domain and Range**
- Domain deals with $x$-values (horizontal)
- Range deals with $y$-values (vertical)
:::
:::danger
**❌ Mistake 2: Using the numerator for domain**
- Only the DENOMINATOR affects domain!
- Numerator $= 0$ is allowed (it just means $y = 0$)
:::
:::danger
**❌ Mistake 3: Forgetting to exclude asymptote values**
- Always state "all reals EXCEPT..."
- Don't just say "all real numbers"
:::
:::danger
**❌ Mistake 4: Wrong asymptote for $y = \frac{ax+b}{cx+d}$**
- Horizontal asymptote is $y = \frac{a}{c}$ (leading coefficients!)
- Not the constant terms!
:::
---
## Slide 15: Practice Problem 1
:::warning
**Find the domain and range:**
$$
f(x) = \frac{5}{x + 1}
$$
:::
**Your turn! Try to solve this before looking at the next slide.**
### Hints:
- What makes the denominator zero?
- What value does $y$ approach as $x \to \pm\infty$?
- This is in the form $y = \frac{a}{x-h} + k$. What are $h$ and $k$?
---
## Slide 16: Solution to Practice Problem 1
:::warning
**Problem:**
$$
f(x) = \frac{5}{x + 1}
$$
:::
### Domain:
- Denominator: $x + 1$
- Set to zero: $x + 1 = 0$
- Solve: $x = -1$
- Vertical asymptote: $x = -1$
:::success
**Domain:** $\{x \mid x \neq -1\}$
:::
### Range:
- Form: $y = \frac{5}{x+1} + 0$
- $k = 0$
- Horizontal asymptote: $y = 0$
- Graph cannot reach $y = 0$
:::success
**Range:** $\{y \mid y \neq 0\}$
:::
---
## Slide 17: Practice Problem 2
:::warning
**Find the domain and range:**
$$
g(x) = \frac{3x - 2}{x - 4}
$$
:::
**Your turn! This one is a bit more challenging.**
### Hints:
- What value of $x$ makes denominator $= 0$?
- For $y = \frac{ax+b}{cx+d}$, horizontal asymptote is $y = \frac{a}{c}$
- What are the leading coefficients $a$ and $c$?
---
## Slide 18: Solution to Practice Problem 2
:::warning
**Problem:**
$$
g(x) = \frac{3x - 2}{x - 4}
$$
:::
### Domain:
- Denominator: $x - 4$
- Set to zero: $x - 4 = 0$
- Solve: $x = 4$
- Vertical asymptote: $x = 4$
:::success
**Domain:** $\{x \mid x \neq 4\}$
:::
### Range:
- Form: $\frac{ax+b}{cx+d}$
- $a = 3$, $c = 1$
- Horizontal asymptote: $y = \frac{3}{1} = 3$
- Graph cannot reach $y = 3$
:::success
**Range:** $\{y \mid y \neq 3\}$
:::
---
## Slide 19: The Big Picture
### Why Domain and Range Matter
- **Real-world applications:** Many situations have natural restrictions (you can't have negative time, etc.)
- **Graphing:** Tells you where the function exists and where it doesn't
- **Problem-solving:** Helps avoid invalid solutions
- **Understanding behavior:** Asymptotes show limiting behavior
### Key Relationships:
| Concept | Controls | Shows |
|---------|----------|-------|
| Vertical Asymptote | Domain | Where function is undefined |
| Horizontal Asymptote | Range | Limiting $y$-value |
---
## Slide 20: Summary
:::success
**Domain:** All $x$-values except where denominator $= 0$
:::
:::success
**Range:** All $y$-values except the horizontal asymptote
:::
### Steps to Remember:
#### For Domain:
1. Find the denominator
2. Set denominator $= 0$
3. Solve for $x$
4. Exclude that $x$-value
#### For Range:
1. Identify function form
2. Find horizontal asymptote
3. Determine excluded $y$-value
4. State "all reals except $y = $ ___"
:::danger
**Don't Forget!**
Always write "except" when stating domain and range for rational functions!
:::
---
## Slide 21: Additional Practice Problems
Try these on your own! Check your answers with your teacher.
### Problem 1:
$$
f(x) = \frac{-2}{x - 7}
$$
### Problem 2:
$$
g(x) = \frac{4}{x + 2} + 3
$$
### Problem 3:
$$
h(x) = \frac{x + 4}{x - 3}
$$
### Problem 4:
$$
y = \frac{5x - 1}{2x + 6}
$$
---
## Slide 22: Answers to Additional Practice
### Problem 1: $f(x) = \frac{-2}{x - 7}$
- **Domain:** All reals except $x = 7$
- **Range:** All reals except $y = 0$
### Problem 2: $g(x) = \frac{4}{x + 2} + 3$
- **Domain:** All reals except $x = -2$
- **Range:** All reals except $y = 3$
### Problem 3: $h(x) = \frac{x + 4}{x - 3}$
- **Domain:** All reals except $x = 3$
- **Range:** All reals except $y = 1$ (because $\frac{1}{1} = 1$)
### Problem 4: $y = \frac{5x - 1}{2x + 6}$
- **Domain:** All reals except $x = -3$ (solve $2x + 6 = 0$)
- **Range:** All reals except $y = \frac{5}{2}$ (because $\frac{5}{2} = 2.5$)
---
## Slide 23: Tips for Success
### ✅ Do:
- Always check the denominator first for domain
- Write "all real numbers except..." clearly
- Remember that vertical asymptotes affect domain
- Remember that horizontal asymptotes affect range
- Practice, practice, practice!
### ❌ Don't:
- Confuse domain and range
- Forget to solve the equation when finding restrictions
- Use the numerator to find domain
- Forget to state the exclusions
- Rush through the problem
:::success
**When in doubt, ask yourself:**
- "What $x$-values break the function?" (Domain)
- "What $y$-values can't be reached?" (Range)
:::
---
## Slide 24: You're Ready!
### Key Takeaways
- **Domain** = $x$-values where function exists (exclude where denominator $= 0$)
- **Range** = $y$-values the function produces (exclude horizontal asymptote)
- **Vertical asymptote** $\Rightarrow$ Domain restriction
- **Horizontal asymptote** $\Rightarrow$ Range restriction
:::success
**Remember:** Mathematics is about understanding patterns. Domain and range help us see the full picture of a function!
**Keep practicing and you'll master this! Good luck! 🚀**
:::
---
## How to Use This Tutorial
### On Screen:
- Read through each section in order
- Work through examples step-by-step
- Try practice problems before looking at solutions
- Use as a study guide
### For HackMD.io:
- This file uses proper LaTeX math syntax with `$...$` for inline and `$$...$$` for display math
- Fractions will render properly with `\frac{numerator}{denominator}`
- Color boxes use HackMD's alert syntax (info, warning, danger, success)
### For Printing:
- Export from HackMD as PDF
- Each section is clearly marked
- Great for creating flashcards
---
*End of Tutorial*
{"title":"Domain & Range V3","contributors":"[{\"id\":\"42414ee5-1acb-4096-af6e-b92dab27f2cb\",\"add\":12510,\"del\":0,\"latestUpdatedAt\":1761479610716}]"}