Problem 67 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Identify the Objective

We need to find the value of \(x\) such that the function value \(f(x)\) equals 1. This means finding the corresponding \(x\) value from the table where \(f(x) = 1\).

Step 2: Examine the Function Table

Look through the \(f(x)\) values in the table to locate where \(f(x) = 1\).

Step 3: Find the x-value

From the table, \(f(x) = 1\) when \(x = 2\). This is because the table shows that when \(x = 2\), \(f(x)\) is indeed equal to 1.

Step 4: Conclude the Solution

State that the value of \(x\) satisfying the equation \(f(x) = 1\) is \(x = 2\).

Key Concepts

Understanding Function Tables

When working with functions in algebra, a function table becomes a very useful tool. It organizes values of the input, represented by \(x\), and their corresponding outputs, represented by \(f(x)\). This table provides a clear visual reference to determine the relationship between inputs and outputs.

Think of it as a two-column list where you have your "\(x\)" values listed in one column and your "\(f(x)\)" values in another.

• The first row includes different values of \(x\).
• The second row contains the resulting \(f(x)\) values for those respective \(x\) values.

For example, in the given exercise, the function table shows various \(x\) values ranging from 0 to 9 and their corresponding \(f(x)\) values. This table is crucial for manually checking calculations and visualizing how changes in \(x\) affect \(f(x)\).

Finding x-values in a Function Table

To solve equations like \(f(x) = 1\), you need to find the specific \(x\) value in the function table. This is a key operation in algebra known as "finding \(x\)-values" that satisfy a given condition.

The process involves searching through the function table for the \(f(x)\) value that meets the specified condition—in this case, where \(f(x) = 1\). This will lead you to the corresponding \(x\) value.

• Look at the list of \(f(x)\) values presented in the table.
• Locate the position where \(f(x)\) exactly equals the desired value, which is 1.
• Identify the associated \(x\) value from the same position.

Once you discover where \(f(x)\) equals 1, you've found the \(x\) value you're searching for.

Solving Equations Using Tables

Finding the value of \(x\) such that \(f(x) = 1\) involves solving an equation using the information provided in a function table. Solving equations this way is direct and involves simple scanning and matching rather than complex computation.

In the given exercise, to solve \(f(x) = 1\), follow these steps:

• Examine each entry in the table under the \(f(x)\) column.
• Identify where exactly \(f(x) = 1\).
• Note the corresponding \(x\) value from the same column of that row.

Here, you find that when \(x = 2\), \(f(x)\) equals 1. This means you've successfully solved the equation. The task is straightforward with a function table, acting like a map guiding you to the solution.

Step by Step Solution Explained

Drawing from the example solution, solving the function \(f(x) = 1\) with a function table can be systematically approached in clear steps. Breaking down each step allows a more comprehensive understanding, especially when learning algebra.

**Step 1:** Identify your goal, which is to find \(x\) when \(f(x) = 1\). This sets a clear target for your search in the function table.

**Step 2:** Assess the function table data. Look through each \(f(x)\) value to pinpoint where it matches the desired outcome, 1, in our case. Visual organization helps easily trace where the function equals the given output.

**Step 3:** Pin down the \(x\)-value. From the matched position of \(f(x) = 1\), inspect the corresponding \(x\) value. This tells you that at this particular \(x\), the function achieves the required value.

**Step 4:** Wrap up the solution by reaffirming that the identified \(x\) is indeed the answer. Here, with \(x = 2\) at \(f(x) = 1\), conclude the task. These structured steps reinforce clarity and confidence in solving similar algebraic problems.