<style>
.slides {
padding: 1in;
max-width: 8in;
margin: 0 auto;
}
</style>
# Domain and Range of Rational Functions
## Algebra 2 - Chapter 7
---
## Part 1: Understanding Domain
---
### What is Domain?
:::info
**Definition**
The **domain** is the set of all possible **input values** ($x$-values) that work in the function.
:::
---
### Think of Domain As:
* All $x$-values you can "plug in"
* All $x$-values that make the function work
* The horizontal extent of the graph
---
### Real-World Example 🎮
Think of a video game character moving left and right.
The **domain** is like the screen boundaries — how far left and right you can go!
---
## Part 2: Understanding Range
---
### What is Range?
:::info
**Definition**
The **range** is the set of all possible **output values** ($y$-values) the function can produce.
:::
---
### Think of Range As:
* All $y$-values the function outputs
* All $y$-values on the graph
* The vertical extent of the graph
---
### Real-World Example 🎮
In the same video game:
The **range** is like how high and low you can jump!
---
## Part 3: Rational Functions
---
### What is a Rational Function?
A rational function has the form:
$$f(x) = \dfrac{p(x)}{q(x)}$$
where:
* $p(x)$ = polynomial on top (numerator)
* $q(x)$ = polynomial on bottom (denominator)
* $q(x) \neq 0$ (denominator cannot be zero!)
---
### The Golden Rule 🚨
:::danger
**You can NEVER divide by zero!**
:::
This means: any $x$-value that makes the denominator zero must be **EXCLUDED** from the domain.
---
### Key Insight 💡
:::success
The **denominator** controls the domain!
The **numerator** does NOT affect domain restrictions.
:::
---
## Part 4: Finding Domain
---
### Step 1: Identify the Denominator
Look at the **bottom** of the fraction.
Example: In $\dfrac{3}{x-5}$, the denominator is $(x-5)$
---
### Step 2: Set Denominator = 0
Set the denominator equal to zero and solve.
Example:
$$x - 5 = 0$$
---
### Step 3: Solve for $x$
Find which $x$-value makes the denominator zero.
Example:
$$x = 5$$
---
### Step 4: Exclude That Value
State the domain excluding that value.
**Answer:** All real numbers except $x = 5$
**Notation:** $\{x \mid x \neq 5\}$
---
## Example 1: Simple Domain
---
### Problem
Find the domain of:
$$f(x) = \dfrac{1}{x}$$
---
### Solution: Step 1
**Identify the denominator:**
The denominator is $x$
---
### Solution: Step 2
**Set denominator = 0:**
$$x = 0$$
---
### Solution: Step 3 & 4
**The problem:** $x = 0$ makes the denominator zero
**Exclude it from the domain**
---
### Answer ✅
:::success
**Domain:** All real numbers except $x = 0$
**Notation:** $\{x \mid x \neq 0\}$
:::
---
## Example 2: Domain with Translation
---
### Problem
Find the domain of:
$$g(x) = \dfrac{3}{x - 2}$$
---
### Solution: Step 1
**Identify the denominator:**
The denominator is $(x - 2)$
---
### Solution: Step 2
**Set denominator = 0:**
$$x - 2 = 0$$
---
### Solution: Step 3
**Solve for $x$:**
$$x = 2$$
---
### Answer ✅
:::success
**Domain:** All real numbers except $x = 2$
**Notation:** $\{x \mid x \neq 2\}$
:::
---
## Example 3: More Complex Domain
---
### Problem
Find the domain of:
$$h(x) = \dfrac{2x + 1}{x - 3}$$
---
### Solution: Step 1
**Identify the denominator:**
The denominator is $(x - 3)$
**Note:** The numerator $(2x + 1)$ doesn't matter for domain!
---
### Solution: Step 2 & 3
**Set denominator = 0:**
$$x - 3 = 0$$
**Solve:**
$$x = 3$$
---
### Answer ✅
:::success
**Domain:** All real numbers except $x = 3$
**Important:** The numerator doesn't affect the domain!
:::
---
## Part 5: Understanding Asymptotes
---
### What is an Asymptote?
:::info
An **asymptote** is a line that the graph approaches but **never touches or crosses**.
:::
---
### Two Types of Asymptotes
1. **Vertical Asymptote** (affects domain)
2. **Horizontal Asymptote** (affects range)
---
### Vertical Asymptote
* A **vertical line**: $x = a$
* Occurs where denominator $= 0$
* Graph shoots up to $+\infty$ or down to $-\infty$
:::success
**Controls DOMAIN**
:::
---
### Horizontal Asymptote
* A **horizontal line**: $y = b$
* What $y$ approaches as $x \to \pm\infty$
* Graph gets closer but never touches
:::success
**Controls RANGE**
:::
---
### The Connection 🔗
:::danger
**Important Rule:**
* Vertical asymptote at $x = a$ → Domain excludes $x = a$
* Horizontal asymptote at $y = b$ → Range excludes $y = b$
:::
---
## Part 6: Finding Range
---
### Step 1: Identify the Form
For $y = \dfrac{a}{x-h} + k$:
The horizontal asymptote is $y = k$
---
### Step 2: Find Horizontal Asymptote
**For** $y = \dfrac{a}{x-h} + k$:
Horizontal asymptote is $y = k$
**For** $y = \dfrac{ax+b}{cx+d}$:
Horizontal asymptote is $y = \dfrac{a}{c}$
---
### Step 3: Determine Excluded $y$-values
The graph can reach **all** $y$-values
**EXCEPT** the horizontal asymptote
---
### Step 4: State the Range
"All real numbers except $y$ = ___"
---
## Example 4: Finding Range
---
### Problem
Find the range of:
$$f(x) = \dfrac{1}{x}$$
---
### Solution: Steps 1 & 2
**This is the parent function form**
As $x$ gets very large (positive or negative), $y$ approaches $0$
**Horizontal asymptote:** $y = 0$
---
### Solution: Steps 3 & 4
The function can output any $y$-value **except** $0$
**Range excludes:** $y = 0$
---
### Answer ✅
:::success
**Range:** All real numbers except $y = 0$
**Notation:** $\{y \mid y \neq 0\}$
:::
---
## Example 5: Range with Translation
---
### Problem
Find the range of:
$$g(x) = \dfrac{-4}{x + 2} - 1$$
---
### Solution: Identify the Form
This is in the form: $y = \dfrac{a}{x-h} + k$
Where:
* $a = -4$
* $h = -2$
* $k = -1$
---
### Solution: Find Asymptote
The horizontal asymptote is $y = k$
$$y = -1$$
The graph shifted **down 1 unit**
---
### Answer ✅
:::success
**Range:** All real numbers except $y = -1$
**Notation:** $\{y \mid y \neq -1\}$
:::
---
## Example 6: Complete Analysis (Part 1)
---
### Problem
Find the **domain** and **range** of:
$$f(x) = \dfrac{2x + 1}{x - 3}$$
---
## Finding the Domain
---
### Step 1: Denominator
The denominator is: $(x - 3)$
---
### Step 2: Set = 0
$$x - 3 = 0$$
---
### Step 3: Solve
$$x = 3$$
---
### Domain Answer ✅
:::success
**Domain:** All reals except $x = 3$
**Vertical asymptote:** $x = 3$
:::
---
## Finding the Range
---
### Step 1: Identify Form
This is: $\dfrac{ax+b}{cx+d}$
Where:
* $a = 2$ (coefficient of $x$ in numerator)
* $c = 1$ (coefficient of $x$ in denominator)
---
### Step 2: Find Asymptote
Horizontal asymptote formula: $y = \dfrac{a}{c}$
$$y = \dfrac{2}{1} = 2$$
---
### Range Answer ✅
:::success
**Range:** All reals except $y = 2$
**Horizontal asymptote:** $y = 2$
:::
---
## Quick Reference Table
---
### Asymptote Formulas
| Function Form | Vertical Asymptote | Horizontal Asymptote |
|---------------|-------------------|---------------------|
| $y = \dfrac{a}{x}$ | $x = 0$ | $y = 0$ |
---
| Function Form | Vertical Asymptote | Horizontal Asymptote |
|---------------|-------------------|---------------------|
| $y = \dfrac{a}{x-h}$ | $x = h$ | $y = 0$ |
---
| Function Form | Vertical Asymptote | Horizontal Asymptote |
|---------------|-------------------|---------------------|
| $y = \dfrac{a}{x-h} + k$ | $x = h$ | $y = k$ |
---
| Function Form | Vertical Asymptote | Horizontal Asymptote |
|---------------|-------------------|---------------------|
| $y = \dfrac{ax+b}{cx+d}$ | $x = -\dfrac{d}{c}$ | $y = \dfrac{a}{c}$ |
---
### Remember 💡
:::success
* Domain excludes **vertical asymptote** values
* Range excludes **horizontal asymptote** values
:::
---
## Common Mistakes (Part 1)
---
### ❌ Mistake 1
**Confusing Domain and Range**
* **Domain** = $x$-values (horizontal)
* **Range** = $y$-values (vertical)
Don't mix them up!
---
### ❌ Mistake 2
**Using the Numerator for Domain**
Only the **DENOMINATOR** affects domain!
If numerator $= 0$, that's okay (it just means $y = 0$)
---
## Common Mistakes (Part 2)
---
### ❌ Mistake 3
**Forgetting to Exclude Values**
Always state: "all reals **EXCEPT**..."
Don't just say "all real numbers"
---
### ❌ Mistake 4
**Wrong Asymptote Formula**
For $y = \dfrac{ax+b}{cx+d}$:
Horizontal asymptote is $y = \dfrac{a}{c}$
Use the **leading coefficients**, not the constants!
---
## Practice Problem 1
---
### Your Turn!
Find the domain and range:
$$f(x) = \dfrac{5}{x + 1}$$
**Try it before looking ahead!**
---
### Hints
* What makes the denominator zero?
* What form is this function in?
* What is the value of $k$?
---
## Solution to Practice 1 (Part 1)
---
### Finding Domain
**Denominator:** $x + 1$
**Set = 0:** $x + 1 = 0$
**Solve:** $x = -1$
---
### Domain Answer ✅
:::success
**Domain:** $\{x \mid x \neq -1\}$
:::
---
## Solution to Practice 1 (Part 2)
---
### Finding Range
**Form:** $y = \dfrac{5}{x+1} + 0$
So $k = 0$
**Horizontal asymptote:** $y = 0$
---
### Range Answer ✅
:::success
**Range:** $\{y \mid y \neq 0\}$
:::
---
## Practice Problem 2
---
### Your Turn!
Find the domain and range:
$$g(x) = \dfrac{3x - 2}{x - 4}$$
**This one is more challenging!**
---
### Hints
* What value makes the denominator zero?
* This is form $\dfrac{ax+b}{cx+d}$
* What are $a$ and $c$?
---
## Solution to Practice 2 (Part 1)
---
### Finding Domain
**Denominator:** $x - 4$
**Set = 0:** $x - 4 = 0$
**Solve:** $x = 4$
---
### Domain Answer ✅
:::success
**Domain:** $\{x \mid x \neq 4\}$
:::
---
## Solution to Practice 2 (Part 2)
---
### Finding Range
**Form:** $\dfrac{ax+b}{cx+d}$ where $a = 3$, $c = 1$
**Horizontal asymptote:** $y = \dfrac{a}{c} = \dfrac{3}{1} = 3$
---
### Range Answer ✅
:::success
**Range:** $\{y \mid y \neq 3\}$
:::
---
## Why This Matters
---
### Real-World Applications
* Many situations have natural restrictions
* You can't have negative time
* Prices must be positive
* Speeds have limits
---
### Mathematical Applications
* **Graphing:** Shows where function exists
* **Problem-solving:** Avoids invalid solutions
* **Understanding behavior:** Shows limits
---
## Summary: Domain
---
### Finding Domain
:::success
**Domain = All $x$-values except where denominator = 0**
:::
**Steps:**
1. Find denominator
2. Set denominator $= 0$
3. Solve for $x$
4. Exclude that value
---
## Summary: Range
---
### Finding Range
:::success
**Range = All $y$-values except horizontal asymptote**
:::
**Steps:**
1. Identify function form
2. Find horizontal asymptote
3. Exclude that value
---
## Quick Memory Aid
---
### Ask Yourself:
**For Domain:**
"What $x$-values **break** the function?"
**For Range:**
"What $y$-values **can't be reached**?"
---
## Additional Practice (Problem 1)
---
$$f(x) = \dfrac{-2}{x - 7}$$
Find domain and range.
---
## Additional Practice (Problem 2)
---
$$g(x) = \dfrac{4}{x + 2} + 3$$
Find domain and range.
---
## Additional Practice (Problem 3)
---
$$h(x) = \dfrac{x + 4}{x - 3}$$
Find domain and range.
---
## Additional Practice (Problem 4)
---
$$y = \dfrac{5x - 1}{2x + 6}$$
Find domain and range.
---
## Answers (Problem 1)
---
$$f(x) = \dfrac{-2}{x - 7}$$
**Domain:** All reals except $x = 7$
**Range:** All reals except $y = 0$
---
## Answers (Problem 2)
---
$$g(x) = \dfrac{4}{x + 2} + 3$$
**Domain:** All reals except $x = -2$
**Range:** All reals except $y = 3$
---
## Answers (Problem 3)
---
$$h(x) = \dfrac{x + 4}{x - 3}$$
**Domain:** All reals except $x = 3$
**Range:** All reals except $y = 1$
(because $\dfrac{1}{1} = 1$)
---
## Answers (Problem 4)
---
$$y = \dfrac{5x - 1}{2x + 6}$$
**Domain:** All reals except $x = -3$
(solve $2x + 6 = 0$)
**Range:** All reals except $y = \dfrac{5}{2} = 2.5$
---
## Tips for Success (Part 1)
---
### ✅ Do This:
* Check the denominator **first**
* Write "all real numbers **except**..."
* Remember vertical asymptotes affect domain
* Remember horizontal asymptotes affect range
---
## Tips for Success (Part 2)
---
### ❌ Don't Do This:
* Don't confuse domain and range
* Don't forget to solve the equation
* Don't use the numerator for domain
* Don't rush!
---
## Final Takeaway
---
:::success
**Domain** = $x$-values where function exists
(exclude where denominator = 0)
:::
---
:::success
**Range** = $y$-values function produces
(exclude horizontal asymptote)
:::
---
:::success
**Vertical asymptote** → Domain restriction
**Horizontal asymptote** → Range restriction
:::
---
## You're Ready! 🎉
---
### Keep Practicing!
Mathematics is about understanding **patterns**.
Domain and range help us see the **full picture** of a function!
---
### Good Luck! 🚀
Remember:
* Take your time
* Show your work
* Ask for help when needed
---
## End of Tutorial
*Created for Algebra 2 Students*
{"title":"Domain and Range V4","description":"Definition","contributors":"[{\"id\":\"42414ee5-1acb-4096-af6e-b92dab27f2cb\",\"add\":12977,\"del\":2,\"latestUpdatedAt\":1761526220752}]"}