34. REASONING Compare and contrast the domain and range of the linear functions described by \( A x+B y=C \) and \( a_{n}=a_{1}+(n-1) d \). 35. CHALLENGE Determine whether each sequence is an arithmetic sequence. Write

### 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.)* --- 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!