Problem 65 - College Algebra

Question

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

Step-by-Step Solution

Answer

3

Step 1: Evaluate the Innermost Function

Start by evaluating the innermost function, which is \( g(6) \). From the given table, we see that when \( x = 6 \), \( g(x) = 7 \). So, \( g(6) = 7 \).

Step 2: Substitute Back into the Outer Function

Now that we have \( g(6) = 7 \), substitute \( 7 \) into the next instance of the function. We now need to find \( g(7) \).

Step 3: Evaluate the Second Function

Using the table, find \( g(7) \). From the table, when \( x = 7 \), \( g(x) = 3 \). Thus, \( g(7) = 3 \).

Step 4: Final Evaluation

The final evaluation of the expression is \( g(g(6)) = g(7) \). Substituting the value we found, \( g(7) = 3 \), the expression evaluates to 3.

Key Concepts

Composite Functions

Composite functions are functions that combine two or more functions into a single expression. The idea is to apply one function to an input and then apply a second function to the result of the first. This is expressed using the notation \((f \circ g)(x)\) or simply \(f(g(x))\). In practice, you must follow a specific order: first evaluate the innermost function and then use its result as the input for the outer function.

Understanding composite functions is crucial because they allow for more complex transformations of inputs compared to individual functions. They can model real-life scenarios where multiple processes occur in stages, each affecting the output of the last.

To approach problems involving composite functions, always start by identifying which function needs to be applied first—this is usually indicated by the function closest to the variable. Evaluate this function, and feed its output into the next function in the sequence.

Table of Values

A table of values is a simple yet powerful tool to map inputs to outputs for specific functions. It provides a clear, organized way to see how a function behaves with different inputs. For example, in the table provided for functions \(f\) and \(g\), each row shows the output of these functions for specific values of \(x\).

Whenever you're dealing with functions expressed through tables, you just need to locate the corresponding output by finding the input value in the table row. This is especially helpful when evaluating nested or composite expressions because it simplifies the search for results. Instead of processing complex calculations, you can directly read off answers, which saves time and reduces errors.

Using a table of values, you can easily trace each step performed in a function evaluation, which is particularly useful when dealing with nested structures as it ensures that no step is skipped or miscalculated.

Nested Functions

Nested functions occur when one function is entirely inside another function. A great example from the problem is \(g(g(6))\), where the output of one function becomes the input of another.

The process of evaluating nested functions follows a step-by-step approach. First, deal with the innermost function. In this case, evaluate \(g(6)\) using the table: when \(x = 6\), \(g(x) = 7\). Substitute this result into the outer function, yielding \(g(7)\).

By breaking down the problem into manageable steps, nested functions become less intimidating. This approach clarifies each mathematical operation, ensuring that you arrive at the correct outcome without losing track of your progress.