Problem 37 - Contemporary Precalculus

Question

Fill the blanks in the given table. In each case the values of the functions \(f\) and \(g\) are given by these tables: $$\begin{array}{|c|c|} \hline x & f(x) \\ \hline 1 & 3 \\ \hline 2 & 5 \\\ \hline 3 & 1 \\ \hline 4 & 2 \\ \hline 5 & 3 \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline t & g(t) \\ \hline 1 & 5 \\ \hline 2 & 4 \\\ \hline 3 & 4 \\ \hline 4 & 3 \\ \hline 5 & 2 \\ \hline \end{array}$$ $$\begin{array}{|c|c|} \hline x & (f \circ f)(x) \\ \hline 1 & \\ \hline 2 & \\\ \hline 3 & 3 \\ \hline 4 & \\ \hline 5 & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Answer: The values of \((f \circ f)(x)\) are: - For \(x=1\), \((f \circ f)(1) = 1\) - For \(x=2\), \((f \circ f)(2) = 3\) - For \(x=3\), \((f \circ f)(3) = 3\) - For \(x=4\), \((f \circ f)(4) = 5\) - For \(x=5\), \((f \circ f)(5) = 1\)

Step 1: Identify the input values of x

The given values of \(x\) in the third table are \(1, 2, 3, 4,\) and \(5\).

Step 2: Find f(x) for each x

Using the first table, find the value of \(f(x)\) for each value of \(x\): - For \(x=1\), \(f(x)=3\) - For \(x=2\), \(f(x)=5\) - For \(x=3\), \(f(x)=1\) - For \(x=4\), \(f(x)=2\) - For \(x=5\), \(f(x)=3\)

Step 3: Find f(f(x)) for each x

Using the values of \(f(x)\) found in Step 2, find the values of \(f(f(x))\) for each value of \(x\) by using the first table again: - For \(x=1\), \(f(f(x))=f(3)=1\) - For \(x=2\), \(f(f(x))=f(5)=3\) - For \(x=3\), \(f(f(x))=f(1)=3\) (already given) - For \(x=4\), \(f(f(x))=f(2)=5\) - For \(x=5\), \(f(f(x))=f(3)=1\)

Step 4: Fill the third table

Now that we have the values of \((f\circ f)(x)\) for each value of x, fill in the third table: $$\begin{array}{|c|c|} \hline x & (f \circ f)(x) \\\ \hline 1 & 1 \\\ \hline 2 & 3 \\\ \hline 3 & 3 \\\ \hline 4 & 5 \\\ \hline 5 & 1 \\\ \hline \end{array}$$

Key Concepts

Table Functions

Table functions are a convenient way to display values and relationships for certain functions without using algebraic expressions. They allow you to quickly see what a function is returning for a given input.
These tables are useful when working with discrete functions, where each input has a particular output defined. They list inputs in one column and their corresponding function outputs in another.
To use a table function:

• Identify the input you need to evaluate.
• Find this input in the table's first column.
• Read across the row to find the corresponding output in the table's second column.

This process helps simplify complex function evaluations and makes understanding function relationships more intuitive by offering a visual representation.

Function Composition

Function composition is a key concept in mathematics where you apply one function to the results of another function. It's often represented as \((f \circ g)(x)\), which means you first apply function \(g\) to \(x\) and then apply function \(f\) to the result of \(g(x)\). This is an essential tool in building more complex functions from simpler ones.
The process allows you to:

• Chain multiple functions together.
• Capture more complex relationships between variables.
• Provide a compact way of expressing nested operations.

Consider the exercise example where we have \(f(f(x))\). This is a composition of the \(f\) function with itself. First, you evaluate \(f(x)\) and then apply \(f\) again to that result. This nested approach provides new interpretations of the function's behavior.

Function Evaluation

Function evaluation involves calculating or finding the value of a function based on a specific input. This process is critical to understanding how a function behaves and what outputs it produces for different inputs.
Here’s how it typically works:

• Identify the input value you need to evaluate in the function.
• Use the function's definition (which could be a formula or a table) to determine the output.
• If composing functions, determine the output of the first function before applying the next.

Consider function \(f(x)\) from the exercise, where you first find \(f(x)\) for each \(x\), and then use these to find \(f(f(x))\), demonstrating the layering typical in function composition. Function evaluation ensures each function's role is clear and distinct, simplifying complex operations into manageable steps.