Problem 60 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

9

Step 1: Find the Inner Function Value

The expression given is \(g(f(5))\). We need to evaluate \(f(5)\) first. From the table, find the value of \(f(x)\) when \(x = 5\). According to the table, \(f(5) = 0\).

Step 2: Use Inner Function Value in Outer Function

Now that we know \(f(5) = 0\), we can substitute this into the outer function \(g(x)\) to find \(g(f(5))\). This means we need to find \(g(0)\).

Step 3: Evaluate the Outer Function

Look at the table to find \(g(x)\) when \(x = 0\). From the table, \(g(0) = 9\). Thus, \(g(f(5)) = g(0) = 9\).

Key Concepts

Table Analysis

When solving problems involving function composition like \(g(f(5))\), the first step is table analysis. The purpose here is to understand the values of the functions \(f(x)\) and \(g(x)\) as provided in the table. Each row in the table represents a combination of \(x\), \(f(x)\), and \(g(x)\).
By reviewing these rows, we can match values of \(x\) to their corresponding function values, which is crucial for evaluating expressions like \(g(f(x))\).

• Locate \(f(x)\) values by finding the specific \(x\) value necessary for your calculation.
• Similarly, locate \(g(x)\) values using the result from the \(f(x)\) evaluation to guide you.

This systematic approach is crucial as it ensures accuracy in determining the correct values for further evaluation in function compositions.

Inner Function Evaluation

The term 'inner function' refers to the function that needs to be evaluated first within a composed function. In our exercise with \(g(f(5))\), the focus is on handling the inner function \(f(5)\) initially.
To evaluate \(f(5)\), simply look at the table and find the row where \(x = 5\). From this row, we observe that \(f(5) = 0\).
Understanding and calculating this inner function correctly is critical because it directly affects the subsequent steps and the final result of the function composition.

• Identify the inner function context, which in our case is \(f(5)\).
• Use the table values to find the exact output for this inner computation.

Correct evaluation at this stage sets the foundation for accurately solving composed functions such as \(g(f(5))\).

Outer Function Evaluation

After determining the value of the inner function, it's time to handle the outer function. In the expression \(g(f(5))\), after calculating \(f(5) = 0\), we substitute this into the outer function \(g\).
The task now is to find \(g(0)\), which involves returning to our trusty table. Locate the row where \(x=0\), and you will see \(g(0) = 9\). Hence, \(g(f(5)) = g(0) = 9\).

• Understand that the outer function takes the result of the inner function as its input.
• Refer back to the table to find the outer function's value for this newly calculated input.

This step finalizes the process of function composition by providing the ultimate result of evaluating the entire function expression.