Problem 58 - 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. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(f(g(8))\)

Step-by-Step Solution

Answer

The value of \(f(g(8))\) is 4.

Step 1: Find g(8)

Look for the row in the table where the first column matches the value 8. In this row, find the corresponding value of \(g(x)\) in the second column, which is \(g(8) = 4\).

Step 2: Find f(g(8)) or f(4)

Use the value from Step 1, which is \(g(8) = 4\), and find the value of \(f(4)\) from the table. Locate the row where the first column is 4 and find the corresponding value in the second column, which gives \(f(4) = 4\).

Step 3: Write the Final Result

With \(f(4) = 4\) found in Step 2, the final result for the expression \(f(g(8))\) is 4.

Key Concepts

Composite Functions

Composite functions might sound complicated, but they're just a way of combining two functions to create a new output. Here's how it works: you have two functions, say \(f(x)\) and \(g(x)\), and you want to apply them one after the other. This is written as \(f(g(x))\).\When evaluating a composite function, you start with the innermost function, in this case \(g(x)\). You take the output of \(g(x)\) and use it as the input for \(f(x)\). It's almost like a two-step process where the output from one function becomes the input to another.\Composite functions are often used in math to perform a series of operations efficiently without needing multiple steps written separately. Remembering the sequence of operations—inner function first, then outer function—can make composite functions much simpler to understand.

Function Tables

Function tables are a fantastic tool for organizing data about functions and their outputs. They help you see at a glance what the outputs (like \(f(x)\) or \(g(x)\)) are for different input values (\(x\)).\In these tables, each row corresponds to a specific input value and defines what that input produces in both functions. For example, if you look at the row where \(x = 8\), you can quickly determine that \(g(8) = 4\) from the table column labeled \(g(x)\).\Using function tables, you avoid recalculating values for each step, which is especially useful when solving composite functions. With practice, you can efficiently read and use these tables to provide accurate results swiftly.

Step by Step Solutions

Understanding mathematical problems is much easier when tackled step by step. Solving problems using a structured approach lets you focus on each part of the problem without feeling overwhelmed. Let's break down the evaluation of the composite function \(f(g(8))\) using steps:\

• Step 1: Start by finding \(g(8)\). Look for the row in the function table where \(x = 8\); here you'll find that \(g(8) = 4\).
• Step 2: Use the output from the first step as the new input for the next function. This means you now need to find \(f(4)\). Check the table where \(x = 4\) and you see \(f(4) = 4\).
• Step 3: Having computed \(f(4)\), you now have your final result. So, the value of \(f(g(8))\) is 4.

Using a step by step solution not only simplifies even the most complex problems but also helps build a foundation of understanding. Investing time in each step ensures that you are sure of the path taken to reach the solution, providing clarity and confidence in your math abilities.