Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Several values of two functions \(f\) and \(g\) are listed in the following tables: $$ \begin{aligned} &\begin{array}{|l|lllll|} \hline \boldsymbol{x} & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 8 & 7 & 6 & 5 & 4 \\ \hline \end{array}\\\ &\begin{array}{|l|lllll|} \hline \boldsymbol{x} & 5 & 6 & 7 & 8 & 9 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & 7 & 8 & 6 & 5 & 4 \\ \hline \end{array} \end{aligned} $$ If possible, find (a) \((f \circ g)(6)\) (b) \((g \circ f)(6)\) (c) \((f \circ f)(6)\) (d) \((g \circ g)(6)\) (e) \((f \circ g)(9)\)

Step-by-Step Solution

Answer

(a) 5, (b) 6, (c) 6, (d) 5, (e) not possible.

Step 1: Understanding composition

The notation \((f \circ g)(x)\) means \(f(g(x))\). This means you first apply the function \(g\) to \(x\), and then apply the function \(f\) to the result of \(g(x)\). Similarly, \((g \circ f)(x)\) means \(g(f(x))\), where you first apply \(f\) and then \(g\). This understanding applies to the other compositions as well.

Step 2: Calculate \((f \circ g)(6)\)

First find \(g(6)\) from the \(g\) table: \(g(6) = 8\). Now find \(f(8)\) using the \(f\) table: \(f(8) = 5\). So, \((f \circ g)(6) = f(g(6)) = f(8) = 5\).

Step 3: Calculate \((g \circ f)(6)\)

Find \(f(6)\) from the \(f\) table: \(f(6) = 7\). Now find \(g(7)\) using the \(g\) table: \(g(7) = 6\). Thus, \((g \circ f)(6) = g(f(6)) = g(7) = 6\).

Step 4: Calculate \((f \circ f)(6)\)

Find \(f(6)\) from the \(f\) table: \(f(6) = 7\). Now find \(f(7)\) since we need \(f(f(6))\): \(f(7) = 6\). Therefore, \((f \circ f)(6) = f(f(6)) = f(7) = 6\).

Step 5: Calculate \((g \circ g)(6)\)

Get \(g(6)\) from the \(g\) table: \(g(6) = 8\). Use this to find \(g(8)\): \(g(8) = 5\). Thus, \((g \circ g)(6) = g(g(6)) = g(8) = 5\).

Step 6: Calculate \((f \circ g)(9)\)

Find \(g(9)\) from the \(g\) table: \(g(9) = 4\). Now determine \(f(4)\) using the \(f\) table: \(f(4)\) is not given in the table, hence \((f \circ g)(9)\) cannot be determined.

Key Concepts

Functions Table

A functions table is a helpful tool in mathematics, especially when dealing with function compositions. It presents values that functions take for specific inputs in a structured way, making it easy to evaluate and understand functions. For instance, in the problem we have two tables for functions \(f(x)\) and \(g(x)\). Each row provides a set of input values and their corresponding output values for the respective functions.

• The top row consists of input values, commonly known as \(x\)-values.
• The following rows show the output or function values, such as \(f(x)\) and \(g(x)\), calculated for each \(x\).

Using functions tables, you can quickly find out what a function's output is for particular input values, a process called function evaluation. This can be extremely useful for solving algebra problems involving multiple steps or operations like composition.

Algebra Problem-solving

In algebra problem-solving, it's essential to break down complex problems into manageable steps. This approach helps in better understanding and solving problems efficiently, as demonstrated in the processed solution of this exercise.
To tackle algebra problems, begin by carefully analyzing what is being asked and identify the knowns and unknowns. In our exercise, we're tasked with finding compositions of functions \(f\) and \(g\) at given values. Solving these requires:

• Understanding function composition;
• Consulting function tables;
• Performing accurate function evaluations.

Staying organized and methodical is key in algebra. Another helpful technique is to use the given tables to simplify any calculations needed to solve the problem. This method enables you to see patterns and relationships between the input and output values across different functions.

Composing Functions

Composing functions involves combining two functions to form a new function, which is a central concept in this exercise. Specifically, when we talk about the composition of functions, we mean operating one function within another. If you have two functions \(f\) and \(g\), the composition \(f \circ g\) is calculated by applying \(g\) first and then applying \(f\) on the result of \(g(x)\). Similarly, \(g \circ f\) means first applying \(f\) and then \(g\).
This process can be visualized or calculated using the function tables, where for example:

• Find \(g(x)\) using the table.
• Use the output of \(g(x)\) as the input for \(f\) using \(f(x)\) table.

Understanding how functions interact through composition is useful for tackling more advanced algebraic challenges and can be applied to many mathematical areas including calculus.

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input. It's fundamental in solving problems involving functions. In our exercise, this involves consulting the function tables, which list the outputs corresponding to different inputs.

• To evaluate \(f(x)\), you'd find \(x\) in the first row of the \(f\) table, then read vertically to the row beneath it.
• Similarly, for \(g(x)\), locate \(x\) in the top row of the \(g\) table and check the corresponding entry below.

Function evaluation is crucial when working with compositions of functions, as each step in a composition may require multiple evaluations to find the final result. It lays the groundwork for understanding how functions change and provide outcomes in various mathematical contexts.