Problem 61 - 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. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(g(f(3))\)

Step-by-Step Solution

Answer

The value of \(g(f(3))\) is 4.

Step 1: Identify f(3)

Look at the table for the value of the function \(f\) when \(x = 3\). In the table, when \(x = 3\), \(f(x) = 8\). Therefore, \(f(3) = 8\).

Step 2: Identify g(f(3))

Now that we know \(f(3) = 8\), we need to find \(g(8)\). Refer back to the table and find the value of \(g(x)\) when \(x = 8\). The table shows \(g(8) = 4\).

Key Concepts

Understanding Table Values

Tables are powerful tools used to organize data in a way that is easy to read and understand. In mathematics, they can help us quickly find the value of a function for a given input by looking at the intersections of rows and columns. Each row represents a unique input, often denoted by \(x\), and the corresponding values of different functions, such as \(f(x)\) or \(g(x)\), fill the columns. For example, in the given table, if you are tasked with finding \(f(3)\), you locate \(x = 3\) in the first column and then move across that row to find \(f(x) = 8\). This approach allows us to efficiently evaluate functions without carrying out an entire calculation process each time. It becomes especially useful in exercises that involve multiple evaluations of different functions.

Exploring Composite Functions

Composite functions might sound complex, but they're actually quite straightforward once you understand the basic concept. A composite function takes the output of one function and uses it as the input for another function. This is often written in the form \(g(f(x))\), which means that you first evaluate \(f(x)\) and then use that result as the input for \(g\). Let's say we have the composite function \(g(f(3))\). We start by finding \(f(3)\) using the table, which we determined to be 8. Next, we take that output and find \(g(8)\), which, according to the table, equals 4. It’s like putting one function inside another, creating a chain of evaluations that leads to our final answer. Understanding composite functions is crucial in calculus and higher-level mathematics because they help in understanding the behavior and properties of more complex equations.

Following a Step-by-Step Solution

Working through a problem step by step is a valuable approach to understanding and solving mathematical problems. Breaking down a problem into smaller, manageable steps can make even the most challenging questions more approachable. Let's go over our solution for \(g(f(3))\) one last time:

• **Step 1:** Find \(f(3)\). Look for \(x=3\) in the table and find that \(f(x)=8\), so \(f(3) = 8\).


• **Step 2:** Use \(f(3)\) as the new input for \(g\). We need to find \(g(8)\). With \(x=8\), the table shows \(g(x)=4\). Thus, \(g(8) = 4\).

Approaching problems in this manner gives clarity to the process and reduces the possibility of missing crucial steps. Step-by-step solutions help in building a systematic method for solving a wide array of function-related problems with confidence.