Answer
**Problem 34:**
- **Linear Function (\( A x + B y = C \)):**
- **Domain:** All real numbers (\( \mathbb{R} \)) if \( B \neq 0 \); otherwise, a single \( x \)-value.
- **Range:** All real numbers (\( \mathbb{R} \)) unless the line is horizontal.
- **Arithmetic Sequence (\( a_{n} = a_{1} + (n-1) d \)):**
- **Domain:** Positive integers (\( \mathbb{N} \)).
- **Range:** A discrete set of values based on the sequence formula.
**Problem 35:**
Cannot determine whether the sequences are arithmetic without the specific sequences provided.
Solution
### Problem 34: REASONING
**Compare and contrast the domain and range of the linear functions described by \( A x + B y = C \) and \( a_{n} = a_{1} + (n-1) d \).**
---
#### **1. Understanding Each Function**
**a. \( A x + B y = C \):**
- **Type of Function:** This is the general form of a linear equation in two variables, representing a straight line on the Cartesian plane.
- **Solving for \( y \):**
\[
B y = -A x + C \\
y = \left(-\frac{A}{B}\right) x + \frac{C}{B}
\]
*(Assuming \( B \neq 0 \))*
- **Interpretation:** This can be seen as a linear function \( y = m x + b \), where:
- **Slope (\( m \))**: \( -\frac{A}{B} \)
- **Y-intercept (\( b \))**: \( \frac{C}{B} \)
**b. \( a_{n} = a_{1} + (n-1) d \):**
- **Type of Function:** This is the formula for the \( n^{th} \) term of an **arithmetic sequence**.
- **Components:**
- \( a_{1} \): First term of the sequence.
- \( d \): Common difference between consecutive terms.
- \( n \): The term number (a positive integer).
---
#### **2. Domain and Range Comparison**
**a. \( A x + B y = C \):**
- **Domain:**
- If viewed as a function \( y(x) \), the domain is **all real numbers** (\( \mathbb{R} \)), provided \( B \neq 0 \).
- **Exception:** If \( B = 0 \), the equation becomes \( A x = C \), which represents a **vertical line**. In this case:
- **Domain:** \( x = \frac{C}{A} \) (a single value)
- **Range:** All real numbers (\( \mathbb{R} \))
- **Range:**
- Generally, **all real numbers** (\( \mathbb{R} \)) for \( y \), unless the line is horizontal (\( A = 0 \)):
- **For \( A = 0 \):**
- **Range:** \( y = \frac{C}{B} \) (a single value)
- **Domain:** All real numbers (\( \mathbb{R} \))
**b. \( a_{n} = a_{1} + (n-1) d \):**
- **Domain:**
- **Positive integers** (\( \mathbb{N} \)) representing the term positions (\( n = 1, 2, 3, \ldots \)).
- **Range:**
- **Discrete set of values** generated by the sequence.
- Each term \( a_{n} \) is a specific real number based on \( n \).
---
#### **3. Key Contrasts and Comparisons**
- **Domain:**
- **Linear Function (\( A x + B y = C \)):**
- Typically **all real numbers** (if \( B \neq 0 \)).
- **Specific cases** (vertical line) have a single \( x \)-value.
- **Arithmetic Sequence (\( a_{n} \)):**
- **Only positive integers**, representing term positions.
- **Range:**
- **Linear Function:**
- **All real numbers**, unless the function is horizontal, limiting \( y \) to a single value.
- **Arithmetic Sequence:**
- **A discrete set** of values, each determined by the formula for \( a_{n} \).
- **Nature of Output:**
- **Linear Function:** Produces a continuous line in the plane.
- **Arithmetic Sequence:** Produces a sequence of distinct, separate points (terms).
- **Type of Function:**
- **Linear Function:** A function of a real variable (\( x \)) mapping to \( y \).
- **Arithmetic Sequence:** A function of a discrete variable (\( n \)) mapping to term values (\( a_{n} \)).
---
### Problem 35: CHALLENGE
**Determine whether each sequence is an arithmetic sequence. Write**
*(Note: The problem statement appears to be incomplete. To provide a comprehensive answer, please supply the specific sequences that need to be evaluated.)*
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If you provide the specific sequences for Problem 35, I’d be happy to help determine whether they are arithmetic sequences and explain the reasoning!
Reviewed and approved by the UpStudy tutoring team
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