Problem 38 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Understand the Problem

We need to find the values of \(x\) for which \(f(x) = 3\). This means we will look at the row of \(f(x)\) and find when it equals 3 and then identify the corresponding \(x\) value.

Step 2: Analyze the Table

Observe the row containing \(f(x)\) values in the table. These are 8, 0, 7, 4, 2, 6, 5, 3, 9, and 1, sequentially corresponding to \(x\) values 0 through 9.

Step 3: Locate \(f(x) = 3\)

Look for the value 3 in the \(f(x)\) row. It appears in the column where \(x = 7\).

Step 4: Identify the Solution

Since \(f(x) = 3\) when \(x = 7\), we conclude that the solution is \(x = 7\).

Key Concepts

Function Evaluation

Function evaluation is the process of determining the output of a function for a given input. In this exercise, we're asked to evaluate the function values from a table, specifically to find where the function \( f(x) \) equals 3. Here, the function \( f \) is defined by a set of specific output values for corresponding input values. Instead of a formula, we're using a table to evaluate the function, which lists the outputs (\( f(x) \)) directly for each input \( x \). This type of evaluation is straightforward: simply locate the desired output in the referred section and find its corresponding input to solve the exercise.

Input-Output Mapping

Input-output mapping is an essential concept when dealing with functions, whether given by a formula or, as in this case, a table. It involves associating each input (in our context, \( x \) values) with exactly one output (\( f(x) \)). Each input value must map to one unique output value. In our given table, input values \( x \) range from 0 to 9, and they're matched to specific outputs, \( f(x) \). The exercise's key task is to understand this mapping to find which input maps to the output of 3. The mapping process simplifies the problem-solving exercise, highlighting which input is linked to our target output.

Table Values

Table values in function solving present a clear and organized way to define function relationships without explicit equations. In our exercise, the table provides a direct representation of function values. Each column in the table corresponds to a specific input-output pair. To find any function value, like \( f(x) = 3 \), you simply scan the \( f(x) \) row to identify the position of the required output. Then, you can find the corresponding \( x \) in the column above. The table concretely shows each pair, making it a practical tool for solving and understanding function-related tasks.

Step by Step Solving

Step by step solving breaks down complex problems into manageable parts, making it easier to comprehend and solve them. Here, it involves the following stages:

• Understand the problem: Identify what you're being asked to find. In this case, we want \( f(x) = 3 \).
• Analyze the data: Look carefully at the table's information, noting each \( f(x) \) value alongside its corresponding \( x \) value.
• Locate the specific value: Find where 3 appears in the \( f(x) \) row.
• Reach a conclusion: Identify the \( x \) value in the same column as the 3, which gives you the solution \( x = 7 \).

Each step ensures clarity and completeness, guiding you systematically to the correct answer.