Problem 47 - Precalculus : Building Concepts and Connections

Question

In Exercises \(41-48,\) use \(f\) and \(g\) given by the following tables of values. $$\begin{array}{ccccc}x & -1 & 0 & 3 & 6 \\\\\hline f(x) & -2 & 3 & 4 & 2\end{array}$$ $$\begin{array}{ccccc}x & -2 & 1 & 2 & 4 \\\\\hline g(x) & 0 & 6 & -2 & 3\end{array}$$ Is \((g \circ f)(0)\) defined? Why or why not?

Step-by-Step Solution

Answer

No, \((g \circ f)(0)\) is not defined because the value of \(g(3)\) is not given in the table.

Step 1: Find the value of \(f(0)\)

From the given table of values for function \(f\), the value of \(f(0)\) is 3.

Step 2: Check if \(g(f(0))\) is defined

Substitute \(f(0) = 3\) into \(g(x)\) to get \(g(3)\). However, the value of \(g(3)\) is not given in the table, so \(g(f(0)) = g(3)\) is not defined.

Step 3: Determine if \((g \circ f)(0)\) is defined

\((g \circ f)(0)\) involves finding \(g(f(0))\). Since \(g(3)\) is not defined from the previous step, \((g \circ f)(0)\) is also not defined.

Key Concepts

Function Tables

Function tables provide a simple way to list values that a function takes for certain inputs. Here, you have two separate tables for functions \( f \) and \( g \). The first table lists values for \( f(x) \), associating each \( x \) value with a corresponding \( f(x) \) value. For example, when \( x = 0 \), \( f(x) = 3 \).
Similarly, the second table shows values for \( g(x) \). These tables help easily access specific function values without needing to solve any equations. Tables save time when working with pre-calculated function outputs, making them a great tool in mathematics.
To use a function table:

• Find the row for the input value \( x \) you are interested in.
• Look across the row to identify the output value of the function.
• Remember, if an \( x \) value is not present in the table, the function is undefined for that input.

For example, you cannot find \( g(3) \) within the given table, meaning it's undefined.

Function Composition

Function composition involves combining two functions in a specific order. Written as \((g \circ f)(x)\), it means applying \( f \) first and then applying \( g \) to the result of \( f \). Think of it as a chain, where the output of one function becomes the input of another.
Consider the problem where you want to find \((g \circ f)(0)\):

• First, calculate \( f(0) \), which, according to the table, is 3.
• Next, we attempt to calculate \( g(3) \). If \( g(3) \) had a defined value, we would use it as the final result.
• If the value is not listed, \( g(3) \) remains undefined, and so does \((g \circ f)(0)\).

This chaining process often reveals how sequential the operation of functions can be, and highlights any missing values that may cause undefined compositions.

Undefined Values

Undefined values in function operations are common when dealing with function tables. An undefined function value occurs when the input does not correspond to a listed output.
In the exercise, because \( g(3) \) isn't available in the table, we encounter an undefined situation. Here's how it breaks down:

• After finding \( f(0) = 3 \), you search for \( g(3) \) in the \( g \) function table.
• \( g(3) \) does not appear in the table, indicating \( g(3) \) is not defined.
• If any part of a function composition involves an undefined result, the entire composition will be undefined.

This understanding is crucial as it teaches you to check for completeness in function tables and alert to missing values that can affect calculations.