Problem 37 - College Algebra

Question

For the following exercises, use the values listed in Table 6 to evaluate or solve. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8 & 0 & 7 & 4 & 2 & 6 & 5 & 3 & 9 & 1 \\\ \hline \end{array} $$ Find \(f(1)\).

Step-by-Step Solution

Answer

The value of \(f(1)\) is 0.

Step 1: Understand the Table

The table provided shows two rows: one for the values of \(x\) and one for the corresponding values of the function \(f(x)\). Each value of \(x\) has a specific function value \(f(x)\) associated with it.

Step 2: Identify the Given Value of x

We need to find the value of \(f(x)\) when \(x = 1\). This means we will look at the column in the table where \(x = 1\).

Step 3: Retrieve the Function Value

In the table, locate the row that shows \(x = 1\). Directly below it in the second row, you will find \(f(1)\). According to the table, \(f(1) = 0\).

Key Concepts

Function Evaluation

Function evaluation is a key concept in algebra that involves finding the output of a function for a specific input. It’s like a machine that transforms an input value into an output value using a specific rule or pattern. In our exercise, this is represented by the table given. Each input, denoted by \(x\), corresponds to an output, \(f(x)\).
To evaluate a function:

• Start by identifying the input value \(x\) you need to evaluate.
• Locate this input in the table and find the corresponding output.
• The output is the function's value for the given input.

Function evaluation helps us understand the behavior of functions and predict outcomes based on specified inputs. In our case, finding \(f(1)\) means we look at where \(x = 1\) in the table, and find the corresponding \(f(x)\) value.

Step-by-Step Solution

Breaking problems into smaller steps is a great way to tackle function evaluation tasks. Here’s the way you simplify and make sense of the problem.

Understand the Table
This is your starting point. Look at the table and notice the rows: the top row shows different \(x\) values, and the bottom row shows corresponding \(f(x)\) values. Each pair aligns vertically with one another.

Identify the Given Value of \(x\)
The problem asks you to find \(f(1)\). Locate \(x = 1\) in the top row of the table.

Retrieve the Function Value
Once \(x = 1\) is found, look directly below in the second row to find \(f(1)\). Transfer that value to your answer confidently.

Interpreting Tables

When working with algebra, interpreting tables is a crucial skill. Tables visually represent relationships between different variables, which makes it easier to understand functions like \(f(x)\).
Here’s how to effectively interpret tables:

• Recognize the structure: Identify which row or column represents the input values (\(x\)) and which represents the output (\(f(x)\)).
• Understand the relationship: Know that each value in the \(x\) row has a direct counterpart in the \(f(x)\) row, which shows the function result.
• Use the table to evaluate functions by matching inputs with their outputs efficiently.

The table in our example lists \(x\) from 0 to 9 and their corresponding \(f(x)\) values. By effectively using the table, we pinpoint results like \(f(1) = 0\) directly and systematically. This skill is invaluable for quickly and accurately solving problems involving functions.