Problem 3 - Precalculus : Building Concepts and Connections

Question

Use the definition of \(f(x)\) as given by the following table. $$\begin{array}{|r|r|} \hline x & f(x) \\ \hline -2 & 5 \\ \hline -1 & 3 \\ \hline 1 & -2 \\ \hline 4 & -1 \\ \hline \end{array}$$ Find \(\left(f \circ f^{-1}\right)(4)\)

Step-by-Step Solution

Answer

The value of \(\left(f \circ f^{-1}\right)(4)\) is 3

Step 1: Find the Inverse Function

We are given the function \(f\) as a table. To find the inverse \(f^{-1}\) we need to switch the role of 'x' and 'f(x)'. This means if (a, b) is on the graph of \(f\) then (b, a) will be on the graph of \(f^{-1}\), thus, the table for \(f^{-1}\) will look like this: \[\begin{array}{|r|r|} \hline f^{-1}(x) & x \ \hline 5 & -2 \ \hline 3 & -1 \ \hline -2 & 1 \ \hline -1 & 4 \ \hline \end{array}\]

Step 2: Evaluate \(f^{-1}(4)\)

Looking at the table for \(f^{-1}\), we see that \(f^{-1}(4)\) equals -1. That means the inverse function \(f^{-1}\) sends '4' to '-1'.

Step 3: Evaluate \(f(f^{-1}(4))\)

Next, we apply \(f\) to the result of \(f^{-1}(4)\), which we found to be -1. Looking at the table for \(f\), we find that \(f(-1)\) equals 3.

Key Concepts

Function Composition

Function composition is an operation that takes two functions and combines them into a single function. In mathematical notation, this is often represented as \( (f \circ g)(x) \), which means you first apply the function \( g \), and then apply the function \( f \) to the result. It's like a two-step process. When given the functions \( f \) and \( g \), creating the composition \( f \circ g \) involves two main steps:

• First, compute \( g(x) \), finding the output of \( g \) when input \( x \) is used.
• Then, take that output value and use it as the input for the function \( f \), resulting in \( f(g(x)) \).

In the example problem, we are asked to find \( (f \circ f^{-1})(4) \). This means you first find the inverse function \( f^{-1}(4) \), and then take that result to find \( f(f^{-1}(4)) \). By doing this, you essentially undo the work of \( f \), arriving back where you started. It's a reflection of how the function and its inverse cancel each other out, akin to taking one step forward and then one step back in a sequence of operations.

Function Tables

Function tables are an organized way to display how each input in a domain is paired with an output in a range. They often help in illustrating the relationship between inputs and outputs for a function and serve as a practical tool for evaluating functions, especially when dealing with simple lists of values.In our exercise, the function \( f \) is defined by a table, which pairs each \( x \) with a corresponding \( f(x) \):

• For \( x = -2 \), we have \( f(x) = 5 \).
• For \( x = -1 \), \( f(x) = 3 \).
• For \( x = 1 \), \( f(x) = -2 \).
• For \( x = 4 \), \( f(x) = -1 \).

To find the inverse function, \( f^{-1} \), simply switch the input and output pairs in the table. This means if the original function maps \( a \to b \), then the inverse function \( f^{-1} \) maps \( b \to a \). With function tables, identifying inverses is straightforward, showing the powerful visualization function tables can provide.

Evaluation of Functions

Evaluating a function involves finding the output value for a given input using the function's rule. Whether provided explicitly as an equation, graphically, or through a table, evaluation requires you to correctly interpret the function's mapping from input to output.In our example with function \( f \), we use a table for evaluation:
1. First, identify the input: for \( f^{-1}(4) \), look at the inverse table where the output is 4 to find the corresponding input, which is -1.
2. Next, evaluate \( f \) at this new input. From the original function table, \( f(-1) = 3 \) gives us our result.
When evaluating composites, it’s crucial to follow the order of operations: apply the inner function first, using its output as the input for the outer function. This careful evaluation helps to ensure the accuracy of results and demonstrates the elegance of functional relationships through composition and inverse evaluation.