Problem 64 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The inverse of the function \(f\) evaluated at \(2\) is \(-1\), \(f^{-1}(2) = -1\).

Step 1: Understanding the function inverse

The inverse of a function \(f^{-1}(x)\) is defined such that if \(f(a) = b\), then \(f^{-1}(b) = a\). In simpler terms, it finds the input that would result in a given output. In this problem, we need to find which input \(x\) gives us an output of \(2\).

Step 2: Looking into the function table

From the table given, look for the output value \(2\) in the column for \(f(x)\). The corresponding \(x\) is the value of \(f^{-1}(2)\). In this case, it can be seen from the table that when \(x\) is \(-1\), \(f(x)\) is \(2\).

Step 3: Writing the final answer

Since \(f(-1) = 2\), this implies that the inverse of the function \(f\) at the point \(2\) is \(-1\), or \(f^{-1}(2) = -1\).

Key Concepts

Function Table

A function table helps us organize and present a mathematical function by listing corresponding inputs and outputs. In the case of the inverse function exercise above, the function table is structured with the input values, represented as \( x \), in one column and the associated output values of the function \( f(x) \) in the adjacent column.

The primary purpose of a function table is to allow us to quickly evaluate a function's outputs given specific inputs. This is particularly helpful in exploring the properties of a function, such as understanding its trend or identifying patterns.

• A, it allows easy comparison between inputs and corresponding outputs.
• B, it gives a clear visualization of the function's behavior.

For example, from the given table in the exercise, we can see the various points and understand how input values like \( -1 \) result in an output of \( 2 \).

Input-Output Relationship

An input-output relationship in mathematics describes how each input value maps to an output value through a function. The function essentially acts as a rule that assigns each input exactly one output.

In the solved exercise, the function \( f \) has its inputs given by the values of \( x \) and outputs provided as \( f(x) \). For example, input \( x = -1 \) produces the output \( f(x) = 2 \).

• The inverse function \( f^{-1} \) reverses this relationship, where given an output, it identifies the initial input.
• This is crucial in finding which input gives a specific output, like when determining that \( f^{-1}(2) = -1 \).

Understanding this relationship enables you to switch between a function and its inverse effectively, helping solve problems requiring tracing back the input from a known output.

Precalculus Concepts

Precalculus provides essential knowledge needed to understand and interpret functions and their inverses, which is critical before advancing to calculus.

In this context, inverse functions are significant as they help us solve equations and model real-world scenarios where a value must be traced back to its origin.

• The notation \( f^{-1} \) represents the inverse function, a fundamental precalculus concept that reverses the action of a function.
• It demands familiarity with identifying and interpreting function tables to easily find inverse values as shown in the exercise, where we determined that \( f^{-1}(2) \) equates to \( -1 \).

Learning precalculus concepts such as inverse functions equips students with analytical tools for later use in calculus while enhancing their mathematical reasoning and problem-solving skills.