Problem 40 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the Inverse Function

The problem requires us to understand that finding \(f^{-1}(x) = 7\) means we need to find the value of \(x\) for which the function \(f(x)\) results in 7.

Step 2: Locate the Value in the Function

Look at the row of \(f(x)\) in the table and locate where the value 7 appears. This will be the power of the inverse function, making the corresponding \(x\) value the answer.

Step 3: Identify the Corresponding x-value

From the table, \(f(x) = 7\) when \(x = 2\). This means that \(f^{-1}(7) = 2\).

Key Concepts

Evaluating Functions

Evaluating functions is a fundamental skill in mathematics. It involves finding the output of a function for a particular input. In simpler terms, if you know the input value, evaluating a function tells you what the output will be. For example, in the context of the function table provided, when we evaluate the function at a specific input (let's say where the function is defined), we use the given function's rule or table to identify what output it produces.

In our example, we were asked to solve for \(f^{-1}(x) = 7\). This requires us to determine what input \(x\) will yield an output of 7. The key is understanding that for a function \(f(x)\), evaluating \(f(x)\) shows you what happens when you substitute \(x\) into the function.

Function Tables

Function tables are a practical way to represent functions, especially when dealing with specific values like in our problem. A function table displays input-output pairs, serving as a quick reference to see how a particular input is transformed into an output.

In our exercise's table, each column in the second row represents an output of the function \(f(x)\) for specific inputs in the first row. This structured format allows you to easily locate outputs and their corresponding inputs. In our case, the table provided a clear pathway to identify \(f(x) = 7\) when \(x = 2\). This straightforward approach often simplifies the process of dealing with functions, especially when dealing with inverse functions or specific values of \(x\).

Step-by-step Solutions

Step-by-step solutions are essential for breaking down complex problems into manageable pieces. This approach helps in understanding not only the "what" but also the "why" behind each action. Let's delve into the steps taken to solve \(f^{-1}(x) = 7\), demonstrating this principle:

• First, identify what the problem is asking. In this case, it's finding the input \(x\) when the output \(f(x)\) is 7.
• Next, use the function table to find where \(7\) appears in the \(f(x)\) row. This locates our desired output in relation to \(x\).
• Identify the corresponding input value. The step-by-step approach made it straightforward to pinpoint that when \(f(x) = 7\), \(x = 2\), hence \(f^{-1}(7) = 2\).

By employing these steps, students can effectively handle similar future problems with confidence and clarity.