Problem 39 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The value of \(f^{-1}(0)\) is 1.

Step 1: Understanding the Goal

To find \(f^{-1}(0)\), we need to determine which \(x\) value from the provided table results in a function value \(f(x) = 0\). In other words, \(f^{-1}(0) = x\) means that when \(f(x) = 0\), the value of \(x\) is the answer.

Step 2: Looking up the Value in the Table

Examine the given table row by row to locate the position where \(f(x) = 0\). In the table, we notice that when \(x = 1\), the value of \(f(x)\) is indeed 0.

Step 3: Concluding the Inverse Discover

Since we found that \(f(1) = 0\), it follows that \(f^{-1}(0) = 1\). Thus, the \(x\)-value corresponding to \(f(x) = 0\) is 1.

Key Concepts

Function Evaluation

Function evaluation involves determining the output of a function for a specific input. In this exercise, you are given a function defined by a table, where each input \( x \) corresponds to a specific output \( f(x) \). Your task is to evaluate or determine the function's inverse, \( f^{-1}(0) \), which means finding an \( x \) that results in \( f(x) = 0 \). Understanding function input and output relationships is vital:

• Each \( x \) in the table represents an input value.
• Each \( f(x) \) is the corresponding output.
• The goal is to find an \( x \) that makes \( f(x) = 0 \).

By assigning specific outputs to inputs, you can retrace this connection to find given inputs from their function outputs during inverse function evaluation. In this case, \( f^{-1}(0) \) means locating the input value whose output is 0 from the table.

Algebra

Algebra plays a crucial role when dealing with functions and their inverses. While evaluating functions and their inverses based on given tables, it's important to apply algebraic thinking:
- Understanding that evaluating an inverse means finding the original input corresponding to a given output.- Recognizing that a function table allows you to quickly reverse input-output pairs.For instance, knowing \( f(1) = 0 \) directly helps us determine \( f^{-1}(0) = 1 \). This algebraic process of inversing involves reversing the relationship defined by the function, converting output back to original inputs.

Step-by-Step Solutions

Step-by-step solutions break down complex problems into manageable parts. They ensure each part of the problem is understood before moving onto the next part. Let's look at how this was done in the given exercise for finding \( f^{-1}(0) \):
1. **Understanding the Task:** The first step is to establish the goal, which is identifying where \( f(x) = 0 \) in the table.2. **Finding the Value:** Review each row of the table systematically to locate where the function value equals 0. Here you see it at \( x = 1 \).3. **Concluding the Solution:** Confirm that the \( x \, \text{value} \) found correctly corresponds to the function output in question, concluding that \( f^{-1}(0) = 1 \).This step-by-step approach helps students better grasp the concept by methodically working through each aspect of the problem. It builds confidence in problem-solving by ensuring all steps are thoroughly comprehensible.