Problem 40 - 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 \(f^{-1}(x)=7\).

Step-by-Step Solution

Answer

\( f^{-1}(7) = 2 \)

Step 1: Understand the Problem

We need to find the inverse function of \( f \) at \( x = 7 \). This means we are looking for the value of \( x \) such that \( f(x) = 7 \). Based on the table, \( f \) assigns a specific value to each \( x \).

Step 2: Locate the Value in the Table

Check the table where \( f(x) = 7 \). We need to find which \( x \) corresponds to this value.

Step 3: Identify the Solution

From the table, find that when \( f(x) = 7 \), the corresponding value of \( x \) is \( 2 \).

Step 4: Write the Solution

Since \( f(2) = 7 \), the inverse function \( f^{-1}(7) \) is \( 2 \).

Key Concepts

Function Evaluation

Function evaluation is a fundamental aspect of understanding functions. It involves determining the output of a function based on a specific input. In simpler terms, it's like asking, "If I put this number into the function, what number will I get out?" For the table provided, function evaluation allows us to understand what the function \( f \) does to each input \( x \).
For instance, if we want to evaluate \( f(3) \), we look at the corresponding value in the table; in this case, \( f(3) = 4 \). This tells us that when \( x = 3 \), the output of the function is \( 4 \).
The process is straightforward:

• Identify the input \( x \) you are interested in.
• Check the table for the corresponding output \( f(x) \).
• Note the function's output for that specific input.

Function evaluation is quite similar to using a vending machine. You select an item (input), and the machine provides you with that item (output). Understanding this concept helps in grasping how changing inputs affect outputs in a function.

Inverse Operations

Inverse operations reverse the effect of the original operation. Think of putting the function into reverse gear; if the function moves forward, the inverse function moves backward. The concept applies similarly to inverse functions, which essentially "undo" the function's operations.
When exploring the problem where \( f^{-1}(x) = 7 \), you are asked to find out for which \( x \) does the function \( f \) give an output of \( 7 \). The inverse function \( f^{-1} \) tells us which input (\( x \)) corresponds to this output (\( 7 \)).
Here's a simple frame to think about inverse operations:

• Forward Function: If \( f(x) = y \), applying \( f \) to \( x \) gives output \( y \).
• Inverse Function: If \( f^{-1}(y) = x \), \( y \) is the result of applying \( f \) to \( x \), hence reversing gives input \( x \).

By finding the inverse, we answer the question "what did we start with?" if we know the end result.

Step by Step Solution

Solving for an inverse function step by step involves a systematic approach.
In this exercise, we were asked to find \( f^{-1}(7) \), meaning we need to identify the input that results in an output of \( 7 \).
Here's how we did it:

• Step 1: Understand the problem by recognizing that finding \( f^{-1}(x) \) when \( x = 7 \) means searching for an \( x \) value where the original function output is \( 7 \).
• Step 2: Check the given table. Look through each \( f(x) \) value listed, searching specifically for \( f(x) = 7 \).
• Step 3: Locate where this occurs in the table. Here, it shows that when \( x = 2 \), \( f(x) = 7 \).
• Step 4: Conclude that \( f^{-1}(7) = 2 \). Write down this solution as a result of your careful examination.

Following each step diligently ensures that you don't miss valuable details in the process of solving similar problems. Moreover, it makes the task straightforward and manageable, helping you connect the dots between an output and its respective input, reversing the original function's path.