Problem 65 - Precalculus : Building Concepts and Connections

Question

Evaluate the given quantity by referring to the function \(f\) given in the following table. $$\begin{array}{cc}x & f(x) \\\\-2 & 1 \\\\-1 & 2 \\\0 & 0 \\\1 & -1 \\\2 & -2\end{array}$$ $$f^{-1}\left(f^{-1}(-2)\right)$$

Step-by-Step Solution

Answer

-1

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

The first step is to find the inverse of the function for the argument -2, that is, \(f^{-1}(-2)\). This means we need to find a \(x\) such that \(f(x) = -2\). Looking at the table, we can see that \(f(2) = -2\). Therefore, \(f^{-1}(-2) = 2\).

Step 2: Find \(f^{-1}\left(f^{-1}(-2)\right)\)

The next step is to find \(f^{-1}\) of the value obtained in the first step, that is, \(f^{-1}(2)\). As per the definition of inverse function, we need to find \(x\) for which \(f(x)=2\). Referring to the table again, it can be seen that \(f(-1)=2\). Hence, \(f^{-1}(2) = -1\).

Step 3: The Final Value

So, the final value of \(f^{-1}\left(f^{-1}(-2)\right)\) is -1.

Key Concepts

Function Evaluation

Function evaluation is essentially about finding the output for a given input value using a function. Imagine you have a machine that, when you input something, gives a specific output. This is how a function works. Given an input value, say \( x \), the function \( f(x) \) gives an output which is determined by the rules or data of the function. For instance, in the table provided, if you input \( x = 0 \), it outputs \( f(0) = 0 \).
Understanding how to properly evaluate a function is crucial:

• Identify the given input value.
• Look up this input in the function's table or use the given function rule.
• Obtain the corresponding output value.

Function evaluation is straightforward yet foundational because it allows you to assess the relationship expressed by the function consistently. For the exercise, the evaluation commenced by finding \( f(2) = -2 \), which is a pivotal step to comprehending how functions work and relate to their inverses.

Inverse Operation

An inverse function is essentially the reverse process of the original function. If a function maps an input to an output, its inverse does the opposite. It is symbolized as \( f^{-1}(x) \). In essence, the purpose of an inverse is to answer the question: "What input leads to a given output?"
The process of deducing an inverse function involves understanding two primary concepts:

• The initial function must be a one-to-one function (bijective).
• Given an output \( y \), find an input \( x \) such that \( f(x) = y \).

In the given exercise, to find \( f^{-1}(-2) \), we identify \( x \) where the function yields \(-2\). Checking the table revealed \( x = 2 \). Continuing, \( f^{-1}(2) \) opens up the inquiry for \( x \) such that \( f(x) = 2 \), which was found to be \( x = -1 \).
Understanding inverse operations allow you to not merely work backwards but also to verify calculations and solve equations where outcomes are known but the initial values are not clear.

Table Interpretation

Interpreting a function table is a skill that makes function evaluation and finding inverses approachable and efficient. A function table represents pairs of input-output values of a function, typically arranged with inputs \( x \) in one column and their corresponding outputs \( f(x) \) in another.
To effectively interpret a table, follow these steps:

• Identify what each column represents, usually \( x \) and \( f(x) \).
• Recognize that each row represents a relationship \( (x, f(x)) \).
• Use the table to find function values and inverses by locating desired entries and corresponding values.

In our essential example, the table was used to efficiently determine \( f^{-1}(-2) \) as \( 2 \), and further to locate \( f^{-1}(2) \) as \(-1\). This underlines the importance of accurately reading and interpreting tables which often present function data concisely. With practice, table interpretation can be used to solve more complex function-related problems swiftly and accurately.