Problem 67 - College Algebra

Question

For the following exercises, use the function \(f\) represented in Table 14 below. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 74 & 28 & 1 & 53 & 56 & 3 & 36 & 45 & 14 & 47 \\ \hline \end{array} $$ Solve \(f(x)=1\)

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the Problem

The problem asks us to find the value of \(x\) for which \(f(x) = 1\) using the given table data.

Step 2: Review the Table

Examine the given table, which lists values of \(x\) and their corresponding \(f(x)\) values. We are interested in rows where \(f(x)\) equals 1.

Step 3: Search for the Function Value Equal to 1

Look through the values in the row labeled \(f(x)\) and find where \(f(x) = 1\).

Step 4: Identify Corresponding x Value

Upon reviewing the table, notice that \(f(x) = 1\) when \(x = 2\).

Step 5: Confirm the Solution

Reverify by checking the table: when \(x = 2\), \(f(x)\) indeed equals 1.

Key Concepts

Function Table

A function table is a tool that helps in analyzing relationships between two variables. In the context of the given exercise, we have a table that aligns each input value of \( x \) to a corresponding output value of \( f(x) \). This form of data presentation makes identifying values and solving function-related questions easier.

Here's how you read and use a function table:

• Identify two rows: one for the input values (\( x \)) and another for the function values (\( f(x) \)).
• Each column corresponds to a pair \((x, f(x))\), showing the relationship for that specific input \( x \).
• Use this organized structure to quickly locate a particular function value and find its corresponding input.

Function tables simplify function evaluation by providing a clear visual representation, reducing the complexity of understanding algebraic relationships.

Algebraic Functions

Algebraic functions involve relationships between variables defined by an equation. An input \( x \) is transformed through an algebraic process into an output \( f(x) \). These functions can be linear, quadratic, polynomial, or even more complex, but all follow a rule or a formula.

To solve for a specific condition of an algebraic function, like finding \( x \) when \( f(x) = 1 \), follow these steps:

• Understand the function's rule or form if provided explicitly.
• Utilize any given structure (e.g., a function table) to identify input-output pairings.
• Recognize patterns or consistent behaviors in function evaluations.

In cases like our exercise, where the algebraic function isn't given as a formula, a function table becomes our main tool for solving.

Problem-Solving Steps

To efficiently solve function-related problems like the one in our exercise, follow a structured approach. This not only ensures clarity but also increases the likelihood of accurately pinpointing solutions.

Here's a breakdown of effective problem-solving steps:

• Understand the Problem: Grasp what the question requests; here, it asks for a specific \( x \) value such that \( f(x) = 1 \).
• Review Given Data: Examine the function table thoroughly, focusing on the row containing \( f(x) \) to find the desired condition.
• Search Methodically: Start from one end of the table and move systematically, checking each \( f(x) \) value until you find the target (1 in our case).
• Verify and Confirm: Once identified, double-check the values by revisiting the table to ensure accuracy.

This step-by-step approach enhances your methodical skills in dealing with algebraic functions and table-based problems, ensuring you don't miss critical details.