Problem 62 - 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. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(f(f(4))\)

Step-by-Step Solution

Answer

\( f(f(4)) = 4 \).

Step 1: Evaluate Inner Function

First, we need to evaluate the inner function of the expression, which is \( f(4) \). Looking at the table, we can see that \( f(4) = 4 \).

Step 2: Evaluate Outer Function

Next, we need to use the result from the inner function to evaluate the outer function. From Step 1, we have \( f(4) = 4 \). So we need to evaluate \( f(4) \) again because the expression is \( f(f(4)) \). Thus, \( f(4) = 4 \).

Key Concepts

Tables in Algebra

Tables in algebra are like a special cheat sheet that helps us understand relationships between numbers. Imagine trying to keep track of how one number changes another without writing it down. A table does this for us. Each row in the table represents a specific value and the corresponding outputs from functions. For example, in our table, the first column has values of \(x\), while the second column shows what happens when you apply the function \(f\) to \(x\), and the third column shows the result of the function \(g\) applied to \(x\).
Using tables, you can easily find values by just looking up the row that corresponds to your \(x\) value. This is incredibly useful for function evaluations, making sure you find the right value quickly and without error. Understanding how to read and use tables is fundamental in algebra, as it forms the basis for more complex problem-solving skills.

Function Composition

In mathematics, function composition is like baking a cake by first mixing the batter and then baking it in the oven. It involves combining two functions to create a new setup where the output of one function becomes the input of another.
In our original problem, we dealt with the composition of the function \(f\) on itself, which is denoted as \(f(f(x))\). Here's what happened in the solution:

• First, we determined the value of \(f(x)\) for a specific \(x\), which was \(4\) in this case.
• Once we found \(f(4)\), which is \(4\), we plugged this 4 back into the function \(f\), leading to \(f(4)\) again.
• This process is similar to layering functions one on top of another to build a final result.

Function composition is a powerful tool in algebra and calculus. It allows for the construction of more complex functions from simpler ones, enhancing your capability to tackle a wide array of mathematical challenges.

Step-by-Step Problem Solving

Approaching math problems with a step-by-step method is like following a recipe. You need to know each step before you can get to the final dish. Solving problems systematically ensures you don’t skip any critical parts and helps to avoid mistakes.
In our example, solving \(f(f(4))\) involved two main steps:

• Evaluate the Inner Function: We first looked at \(f(4)\), using the table to see that \(f(4) = 4\).
• Evaluate the Outer Function: With \(f(4)\) known, we substituted \(4\) into \(f(x)\), evaluating \(f(4)\) again, resulting in \(4\).

By carefully executing each step, you not only solve the isolated problem but also gain confidence in handling similar tasks in the future. Always remember, clear steps are key to mastering mathematical concepts and avoiding errors.