Problem 64 - 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. $$ \begin{array}{|c|c|c|} \hline x & f(x) & g(x) \\ \hline 0 & 7 & 9 \\ \hline 1 & 6 & 5 \\ \hline 2 & 5 & 6 \\ \hline 3 & 8 & 2 \\ \hline 4 & 4 & 1 \\ \hline 5 & 0 & 8 \\ \hline 6 & 2 & 7 \\ \hline 7 & 1 & 3 \\ \hline 8 & 9 & 4 \\ \hline 9 & 3 & 0 \\ \hline \end{array} $$ \(g(g(2))\)

Step-by-Step Solution

Answer

The value of \( g(g(2)) \) is 7.

Step 1: Find g(2) from the table

Look up the value of \( g(2) \) from the table. The table indicates that when \( x = 2 \), \( g(x) = 6 \).

Step 2: Use the result from Step 1 to find g(g(2))

Now that you know that \( g(2) = 6 \), you must find \( g(6) \) using the table. For \( x = 6 \), the table shows \( g(6) = 7 \).

Key Concepts

Function Evaluation

Function evaluation entails determining the output of a function for a specific input. In simpler terms, it's like figuring out what a machine (or function) does when you feed it a certain number (or input). Each function is like a rule or a set of instructions that tells you what to do with the input value to get the output.

Let's take the function \( g(x) \) from our exercise as an example. If you're given \( x = 2 \) and you need to find \( g(2) \), you would use the rule that the function represents, which in this case is the table's value for \( x = 2 \). Here, the lookup shows us that \( g(2) = 6 \).

Function evaluation is foundational in mathematics because it applies to numerous situations. You will often find yourself evaluating functions as part of solving equations, analyzing graphs, or even in data science. It teaches the importance of following steps and using predefined information, much like using a recipe or following assembly instructions.

Tables of Values

Tables of values are simplified formats to display function outputs for a list of input values. They provide a quick way to see patterns, understand function behavior, and make the process of function evaluation straightforward.

In our exercise with the functions \( f(x) \) and \( g(x) \), the table lists values of \( x \) and corresponding outputs for both functions. This allows us to swiftly find function values.

• For example, to find \( g(2) \), you locate the row where \( x = 2 \) and see that \( g(x) = 6 \).
• To find another value, like \( g(6) \), you would find the row for \( x = 6 \), where the table indicates \( g(x) = 7 \).

Tables of values are incredibly useful in both educational and real-world contexts. They help organize information to make it visually friendly and reduce process errors when working with multiple data sets.

Composition of Functions

The composition of functions is a powerful concept in mathematics that involves combining two or more functions in a certain order. The idea is to use the output of one function as the input for the next function.

In the exercise provided, you were asked to find \( g(g(2)) \). This signifies a composite function where you first find \( g(2) \), and then use that result as the new input for \( g(x) \).

The steps involved answer this specific composition:

• First, evaluate \( g(2) \), which equals 6, as found directly from the table.
  
• Secondly, use this result to find \( g(6) \), where the table indicates \( g(6) = 7 \).

Understanding composition of functions is crucial for higher math as it lays the groundwork for more complex topics, like calculus. It shows how functions can be linked and work together, exposing deeper patterns and insights in mathematical relationships.