Problem 64 - College Algebra

Question

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 3 to evaluate each expression. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline {x} & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline {f(x)} & {7} & {6} & {5} & {8} & {4} & {0} & {2} & {1} & {9} & {3} \\ \hline {g(x)} & {9} & {5} & {6} & {2} & {1} & {8} & {7} & {3} & {4} & {0}\\\ \hline \end{array}$$ $$g(g(2))$$

Step-by-Step Solution

Answer

7

Step 1: Identify g(2)

From the table, we find the value of \( g(2) \). Looking at the row for \( g(x) \), when \( x = 2 \), \( g(x) = 6 \). So, \( g(2) = 6 \).

Step 2: Identify g(g(2))

We found that \( g(2) = 6 \) in the previous step. Now, we need to find \( g(6) \). Referring back to the table: For \( x = 6 \), \( g(x) = 7 \). Therefore, \( g(6) = 7 \).

Step 3: Evaluate g(g(2))

Based on the previous steps, \( g(g(2)) = g(6) = 7 \). Thus, the final value for \( g(g(2)) \) is 7.

Key Concepts

Evaluating Functions

In order to evaluate functions, we first need to understand the concept of an input and output within a function, denoted commonly as \( f(x) \) or \( g(x) \). Here, \( x \) represents the input value that you provide to the function, and the output is the value that you receive after the function processes this input. The process of evaluating a function involves substituting the value of \( x \) into the function and then calculating the output. For instance, if you need to evaluate \( g(2) \), you start by plugging \( 2 \) into the function \( g \) and determine the corresponding output from a given set of values or a function rule.

In our specific exercise, the goal is to evaluate \( g(g(2)) \). This double evaluation means that you first need to find \( g(2) \), which provides you with the input for the next evaluation. This is a typical case of composing functions where the output of one function acts as the input for another.

Function Tables

Function tables are a practical tool for organizing and visualizing the relationships between inputs and outputs of a function. They provide a straightforward way to identify the output values for given inputs, especially when dealing with discrete values such as integers.

In the provided problem, we're given a function table for \( f \) and \( g \). Each row in the table corresponds to a function, with the top row listing potential input values (\( x \)) and subsequent rows detailing the output values for \( f \) and \( g \).

Using this table allows us to easily find values such as \( g(2) = 6 \), because we can simply look up the input in the first row and trace down the column to the \( g(x) \) row. These tables are particularly helpful when working through each step of an evaluation or composition, as they ensure that no steps are skipped and every calculation is accurate.

Iterative Function Application

Iterative function application refers to the process where the output of a function becomes the input of the same or another function. This process requires careful attention to detail, as each step provides the foundation for the next.

In the scenario \( g(g(2)) \), the evaluation begins with finding \( g(2) \), yielding an output of \( 6 \). This \( 6 \) is then immediately used as the input for the subsequent application of \( g \); therefore, the next step is to evaluate \( g(6) \). Through this iterative method, you are able to apply the function repeatedly until you reach the final solution.

Understanding this iterative process is crucial for students, as it builds a foundation for more complex computations involving nested functions. This method is frequently seen in advanced mathematics and computer science, where functions are often required to process their own results multiple times to generate a final output.