Problem 96 - College Algebra with Modeling and Visualization

Question

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ \left(g^{-1} \circ g^{-1}\right)(2) $$

Step-by-Step Solution

Answer

The result is 2.

Step 1: Understanding the Composition

To solve the problem \(ig(g^{-1} \circ g^{-1}\big)(2)\), you need to first find \(g^{-1}(2)\). This means finding the value of \(x\) such that \(g(x) = 2\).

Step 2: Finding \(g^{-1}(2)\)

Examine the second table to find the value of \(x\) that satisfies \(g(x) = 2\). From the table, when \(x = 1\), \(g(x) = 2\). Therefore, \(g^{-1}(2) = 1\).

Step 3: Applying \(g^{-1}\) Again

Since \(g^{-1}(2) = 1\), you now need to evaluate \(g^{-1}(1)\). This means finding the value of \(x\) such that \(g(x) = 1\). Looking in the table, when \(x = 2\), \(g(x) = 1\). Hence, \(g^{-1}(1) = 2\).

Step 4: Final Evaluation

Combine the results from the previous steps. Since \(g^{-1}(2) = 1\) and \(g^{-1}(1) = 2\), the final answer for \(\big(g^{-1} \circ g^{-1}\big)(2)\) is 2.

Key Concepts

Function Composition

Function composition involves combining two functions so that the output of one function becomes the input of the other. This is often denoted as \((f \circ g)(x)\), meaning you first apply function \(g\) to \(x\), and then function \(f\) to the result of \(g(x)\). In our specific problem, we are dealing with the composition of the inverse functions, specifically \(g^{-1} \circ g^{-1}\). This notation means we will first find the inverse of \(g(x)\) for a specific value, and then apply the inverse function again using the first result as the input. This double operation can be thought of as applying the inverse operation twice in sequence.

Table Evaluation

To solve problems involving inverse functions efficiently, table evaluation becomes crucial. Tables give a quick snapshot of function inputs and outputs.
For instance, in the first table given, finding \(f(x)\) for a particular \(x\) is straightforward. You just locate \(x\) in the header row and read down the column to find the output. In the problem at hand, we need to look at the second table to find \(g^{-1}(y)\), which is the input \(x\) that corresponds to an output \(y\) in \(g(x)\).
This is done by scanning the row of \(g(x)\) until we find the value \(2\) and \(1\), and then moving upwards to find the corresponding \(x\) values, which are the outputs for the inverse functions we need.

Algebra Problem Solving

In algebra problem solving, understanding inverse functions and how to manipulate them is essential. In many algebra problems, we face situations where we need to reverse processes. With our specific example, the task was to reverse the function \(g(x)\) twice. This requires understanding that each inverse \(g^{-1}\) undoes the operation of its corresponding \(g(x)\).
One useful approach is to clearly identify what is known and what needs to be found.

• First, identify the target value \(y\) (in this case, 2) and use the table to find the corresponding \(x\) for \(g^{-1}\).
• Secondly, use this result as your new target for a second evaluation (finding \(g^{-1}(1)\)).

Applying these steps one by one with clear logical reasoning will lead you to the correct answer.