Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Several values of two functions \(T\) and \(S\) are listed in the following tables: $$ \begin{aligned} &\begin{array}{|l|lllll|} \hline \boldsymbol{t} & 0 & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{T}(\boldsymbol{t}) & 2 & 3 & 1 & 0 & 5 \\ \hline \end{array}\\\ &\begin{array}{|l|lllll|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{S}(\boldsymbol{x}) & 1 & 0 & 3 & 2 & 5 \\ \hline \end{array} \end{aligned} $$ If possible, find (a) \((T \circ S)(1)\) (b) \((S \circ T)(1)\) (c) \((T \circ T)(1)\) (d) \((S \circ S)(1)\) (e) \((T \circ S)(4)\)

Step-by-Step Solution

Answer

(a) 2, (b) 2, (c) 0, (d) 1, (e) Not possible.

Step 1: Understanding composition of functions

The notation \((T \circ S)(x)\) means you first apply \(S\) to \(x\), then apply \(T\) to the result. Similarly, \((S \circ T)(x)\) means you first apply \(T\) to \(x\), then apply \(S\) to the result.

Step 2: Calculating \((T \circ S)(1)\)

First find \(S(1)\), which is the value corresponding to \(1\) in the \(S\) table. \(S(1) = 0\). Next, find \(T(0)\) because \((T \circ S)(1) = T(S(1))\). From the \(T\) table, \(T(0) = 2\). Thus, \((T \circ S)(1) = 2\).

Step 3: Calculating \((S \circ T)(1)\)

First find \(T(1)\), which is the value corresponding to \(1\) in the \(T\) table. \(T(1) = 3\). Next, find \(S(3)\) because \((S \circ T)(1) = S(T(1))\). From the \(S\) table, \(S(3) = 2\). Thus, \((S \circ T)(1) = 2\).

Step 4: Calculating \((T \circ T)(1)\)

First find \(T(1)\), which is the value corresponding to \(1\) in the \(T\) table. \(T(1) = 3\). Next, find \(T(3)\) because \((T \circ T)(1) = T(T(1))\). From the \(T\) table, \(T(3) = 0\). Thus, \((T \circ T)(1) = 0\).

Step 5: Calculating \((S \circ S)(1)\)

First find \(S(1)\), which is the value corresponding to \(1\) in the \(S\) table. \(S(1) = 0\). Next, find \(S(0)\) because \((S \circ S)(1) = S(S(1))\). From the \(S\) table, \(S(0) = 1\). Thus, \((S \circ S)(1) = 1\).

Step 6: Calculating \((T \circ S)(4)\)

First find \(S(4)\), which is the value corresponding to \(4\) in the \(S\) table. \(S(4) = 5\). However, the value 5 is not present in the \(t\) values for \(T\), meaning \(T(5)\) is undefined. Therefore, \((T \circ S)(4)\) is not possible to find.

Key Concepts

Table of Functions

A table of functions is a handy tool that displays the input-output relationship of a function. Here, two tables are used, one for the function \( T \) and another for \( S \).
Each row in a table lists a set of inputs and their corresponding output values.
For example, in the table for function \( T \), the input \( t = 0 \) produces an output \( T(t) = 2 \). Similarly, the entry \( x = 1 \), in the table for function \( S \), gives the result \( S(x) = 0 \).
This arrangement makes it easy to locate the output for any given input by simply reading across the table.

Function Evaluation

Function evaluation is about finding the value of a function at a specific input.
In this exercise, we evaluated several compositions such as \((T \circ S)(1)\). To perform this, you begin by finding the value of the innermost function \( S(1) \). This value then becomes the input for the next function \( T \).
In some cases, this means exchanging multiple values from one table to another until reaching the final outcome. This methodical approach helps ensure accurate results.
Understanding how to read and use function tables is crucial for accurate function evaluation.

Mathematics Problem Solving

Mathematics problem-solving often involves breaking down complex tasks into smaller, manageable steps.
For function compositions, this means applying each function one step at a time, starting with the innermost function.
Once you know the value of the innermost function, you proceed to the next, progressively building until you reach the final result.
For example, solving \((S \circ T)(1)\) involves evaluating \( T(1)\) first, followed by evaluating \( S \) at the resulting value. These steps highlight the importance of procedural thinking, common in mathematics problem-solving.

Undefined Values in Functions

Undefined values in functions occur when there is no corresponding output in the function table for a given input.
In the problem, when trying to find \((T \circ S)(4)\), the issue arises because \( S(4) = 5 \), but there is no \( T(5) \) in the table.
This means that the composition \((T \circ S)(4)\) is not possible, as \( T \) does not have a value for the input 5.
Understanding when a function's value is undefined is key in mathematics, ensuring results are accurate and errors are avoided. It is an important concept, particularly in real-world applications where input values might not always map to outputs cleanly.