Problem 44 - Precalculus : Building Concepts and Connections

Question

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Evaluate \((f \circ g)(4)\)

Step-by-Step Solution

Answer

Therefore, the value of \((f \circ g)(4)\) = 4

Step 1: Compute g(4)

Firstly, look into the table for the function \(g(x)\) and find the corresponding value for when \(x = 4\). From the table it can be seen that \(g(4) = 3\).

Step 2: Compute f(g(4))

Then, using the output from \(g(x)\) as the input for \(f(x)\), find the corresponding value for when \(x = 3\). From the table it can be seen that \(f(3) = 4\). Thus, \(f(g(4)) = f(3) = 4\).

Key Concepts

Function Evaluation

Function evaluation involves plugging a specific value into a function to find the corresponding output. Imagine you have a machine (the function), and you insert a particular piece (the input or value), and out comes a unique result (the output). This process is straightforward with numbers and functions.

• The input value is substituted into the function's equation or checked in a table.
• The result is the function's output corresponding to that input.

For instance, in our exercise, we evaluated the functions given in table form. To find \(g(4)\), we simply looked into the table for \(g(x)\) and found the output. The value given was \(3\), showing the output produced when \(x = 4\) for the function \(g(x)\). Function evaluation helps us understand how the function behaves across different inputs.

Mathematical Tables

Mathematical tables are tools used to simplify function evaluation, especially when dealing with discrete functions. They provide a quick reference for results without calculating them each time.Using tables is a bit like using a handy cheat sheet:

• They list inputs and corresponding outputs for functions.
• Tables are particularly useful when functions are too complex or cumbersome for quick calculations.
• These tables need to be carefully organized, ensuring each input matches its proper output.

In the exercise, tables were provided for functions \(f(x)\) and \(g(x)\). When evaluating \(f(g(4))\), we first found \(g(4)\) using its table, and then used the result as input in the \(f(x)\) table. The tables essentially guided us through the process, reducing calculation errors and making composition straightforward.

Nested Functions

Nested functions, also known as composite functions, involve using the output of one function as the input for another. It's like peeling layers of an onion, revealing the next step hidden within the previous one.Here's how they work:

• The output of the first function becomes the input for the next.
• Nesting functions requires you to evaluate them in the correct order, often guided by parentheses.
• Commonly written as \((f \circ g)(x)\), it means "first apply \(g(x)\), then apply \(f\) to the result".

In our example, we evaluated \((f \circ g)(4)\). First, \(g(4)\) was calculated to be \(3\). Next, this value was used as the input for \(f(x)\), resulting in \(f(3) = 4\). By understanding the order and the steps, once complicated nested functions become highly manageable!