Problem 65 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value of \(g(g(6))\) is 3.

Step 1: Identify Inner Function Values

To evaluate the nested function expression, start by identifying the value of the innermost function, \(g(6)\). From the table, when \(x = 6\), \(g(6) = 7\).

Step 2: Evaluate Outer Function Using Inner Function Result

Now, using the result from the inner function, evaluate the outer function \(g\) at \(x = 7\). From the table, \(g(7) = 3\).

Key Concepts

Function Evaluation

In mathematics, function evaluation is the process of determining the output of a function for a given input. It’s a fundamental concept that helps us understand the relationship between variables. When we evaluate a function like \(g(x)\), we substitute the value of \(x\) into the function and then calculate or look up the resulting value of \(g(x)\).To illustrate: imagine a function \(f(x)\) given in a table. Each value of \(x\) corresponds to an output, \(f(x)\). When you evaluate the function at a specific point, you simply find \(x\) in the table and note the corresponding value. This output is the result of the function evaluation.For example, if you need to find \(f(2)\), you would:

• Look for 2 in the \(x\)-row of the table.
• Locate the number directly beneath it in the \(f(x)\)-row.
• This number is the value of \(f(2)\).

Understanding this process allows you to handle more complex problems with confidence, such as nested function evaluations, by breaking them down into simpler steps.

Table Interpretation

Interpreting a table for function values is a skill that becomes very useful, especially in algebra and calculus. A table organizes data in a way that makes it easy to read and understand the relationship between variables. Every function has an input (independent variable) and an output (dependent variable).Take the given table of values for \(f(x)\) and \(g(x)\). The top row represents possible inputs (\(x\) values). The rows beneath show the corresponding outputs for \(f(x)\) and \(g(x)\) respectively. This arrangement helps us to quickly determine the result of evaluating a function at specific points.Here’s how to interpret such a table:

• Identify the column representing the input value you are interested in.
• Check the outputs in the corresponding rows beneath this column to find \(f(x)\) and \(g(x)\).

Understanding this interpretation allows one to efficiently manage and solve problems involving multiple functions using provided values from tables or charts.

Nested Functions

Nested functions involve placing one function inside another. This occurs often in composite functions where the output of one function becomes the input for another function. It might seem complicated at first, but tackling it step-by-step can simplify the process significantly.For example, consider the expression \(g(g(6))\). Here, you have one function nested inside another, both given by the same function \(g\). To solve it:

• Start with the innermost function: evaluate \(g(6)\). From the given table, \(g(6) = 7\).
• Use this result as the input for the outer function: substitute \(7\) into \(g(x)\). Subsequently, \(g(7) = 3\).

This sequential process—evaluating the inner function and then using its output for the outer function—ensures accuracy. Understanding nested functions is essential for advanced topics in mathematics as it forms the backbone of many operations in calculus and real-world applications where multiple processes interact in stages.