Several values of two functions T and S are listed in the...

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

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

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Key Concepts

Function Composition

This is the process of applying one function to the result of another. In expressions like (T ? S)(x), the function S is applied first to x, and then T is applied to the result of S(x). It is a fundamental concept in mathematics where functions are combined to form new functions, and the order of application is very important.

Interpreting Function Tables

When functions are defined by tables, each table entry indicates a mapping from an input (often listed in one row or column) to its corresponding output (listed in another row or column). Reading these tables correctly involves identifying the input value of interest and then finding its corresponding output. This skill is crucial for evaluating functions at specific points and for carrying out operations such as composition.

Domain and Range Considerations

In the context of compositions, it is essential to ensure that the output from the first function (the inner function) lies within the domain of the second function (the outer function). This check guarantees that the composition is well-defined. Understanding the domain and range of the functions involved is fundamental to determining whether a composite function can be evaluated for a given input.

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