Problem 63 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value of \( f(f(1)) \) is 2.

Step 1: Extract Inner Function Value

First, identify the value of the inner function, which is \( f(1) \). From the table, \( f(1) = 6 \).

Step 2: Evaluate Outer Function

Next, use the value from the inner function, \( f(1) = 6 \), to find \( f(f(1)) = f(6) \). From the table, \( f(6) = 2 \).

Key Concepts

Understanding Function Evaluation

Function evaluation is the process of determining the output of a function for a given input. In other words, it’s like asking, "What do we get if we plug in a certain number into our function?"
For example, when we have a function \( f(x) \), we want to find out what value comes out when we put a certain \( x \) value into it. If given \( x = 1 \), function evaluation would mean figuring out what \( f(1) \) equals to by looking it up in the table.
In our exercise, evaluating \( f(1) \) means checking the table where \( x = 1 \) and finding that \( f(1) = 6 \). This understanding of function evaluation is crucial, as it builds the groundwork for evaluating more complex expressions like \( f(f(x)) \).

Step-by-Step Solution Explained

A step-by-step solution breaks down the problem-solving process into smaller, manageable parts. It makes understanding easier by tackling one part of the problem at a time. Let's break down how this approach works using the given exercise.

• Step 1: Find the Inner Function's Value
  In the expression \( f(f(1)) \), we first evaluate the innermost part, which is \( f(1) \). From the table, we know that when \( x = 1 \), \( f(1) = 6 \).
• Step 2: Use the Inner Function's Result
  Now, we take the result from step 1 (which is 6) and evaluate the function again. We need to find \( f(6) \). Look at the table to find what \( f(6) \) equals to, and we see that \( f(6) = 2 \).
  By following these steps, we see that \( f(f(1)) = 2 \).
  Breaking down each part highlights the logical flow and works for complexity step by step, ensuring accuracy and clarity.

Function Values Table as a Tool

A function values table is a handy tool that shows the direct relationship between different \( x \) values and their corresponding function outputs. It acts as a quick reference point to find function values without needing to compute them every time.
In this exercise, we have two functions, \( f \) and \( g \), and their corresponding values for \( x \) ranging from 0 to 9. Here’s how to use the table effectively:

• Locate the correct row for the function you are evaluating, either \( f \) or \( g \).
• Find the column that corresponds to your specific \( x \) value.
• Read off the function output directly where these intersect.

For instance, if you want to know \( f(4) \), look for the position where row \( f(x) \) and column \( x = 4 \) meet. You’ll quickly find that \( f(4) = 4 \).
Using a function values table not only speeds up the problem-solving process but also reduces the potential for errors, ensuring that every function evaluation is accurate and efficient.