Problem 62 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Find initial function value

First, determine the value of \( f(4) \). Looking at the row corresponding to \( f(x) \) in the table, when \( x = 4 \), we find \( f(4) = 4 \).

Step 2: Evaluate inner function

With \( f(4) = 4 \), we need to evaluate the inner function in the expression \( f(f(4)) \), which means we require \( f(4) \) again. From the table, this gives us \( f(4) = 4 \) once more.

Step 3: Evaluate composed function

Finally, use the initial and derived value to find \( f(f(4)) \). Since \( f(4) = 4 \) from the earlier steps, evaluate \( f(f(4)) = f(4) = 4 \).

Key Concepts

Function Tables

Function tables are a fundamental tool in mathematics, used to represent the values of a function for specific inputs. Imagine them as a checklist that helps us evaluate the outcomes of functions for particular inputs. In our example, we have a table that provides the outputs for functions \( f(x) \) and \( g(x) \) for inputs from 0 to 9.
These tables simplify the process of finding function values by eliminating the need for complex calculations.Using these tables is straightforward:

• Locate the row of the required function (either \( f(x) \) or \( g(x) \)).
• Find the column headed by the input value (\( x \)).
• The intersection of the row and column gives you the output.

In our specific example, to find \( f(4) \), we look at the row labeled \( f(x) \) and the column for \( x = 4 \). This intersection yields the value \( f(4) = 4 \). Function tables are practical and objective, making them ideal for quickly finding values without having to work through each calculation step-by-step.

Evaluating Functions

Evaluating a function means finding the output when we substitute a given input into the function. This process is like using a vending machine: you input a number, and out comes the function's output. The function table helps streamline this by showing values at a glance.Here's how to evaluate a function:

• Identify the function you need to evaluate. It could be \( f(x) \) or \( g(x) \).
• Use the table to find the output for the specific input, such as \( f(4) \) or \( g(4) \).
• Write down and use this output in further calculations or expressions.

In our example, to evaluate \( f(4) \), you check the function table and see that the output is 4. So, if you start with \( f(x) \), you simply replace the placeholder \( x \) with the given number and find the corresponding outcome. Evaluating functions becomes very efficient using well-organized function tables, as they directly supply these pertinent values.

Composed Functions

Composed functions involve the integration of one function with another. Essentially, the output of one function becomes the input for another. It’s like nesting one action inside another, similar to applying two chained steps to achieve a final outcome. This concept is essential when dealing with expressions such as \( f(f(x)) \) or \( g(f(x)) \).To evaluate composed functions:

• Start by finding the inner function's output using the function table.
• Take that output and use it as the input for the outer function.
• Evaluate the outer function using this new input value.

In our exercise, evaluating \( f(f(4)) \) involves two steps. First, determine \( f(4) \), which equals 4. Next, use this result as the input for \( f \) once again, leading to \( f(4) = 4 \). Thus, \( f(f(4)) = 4 \). This technique, while sometimes tricky, becomes manageable with consistent practice and a clear function table, facilitating proper handling of both simple and complex composed functions.