Problem 91 - College Algebra with Modeling and Visualization

Question

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ f^{-1}(3) $$

Step-by-Step Solution

Answer

\( f^{-1}(3) = 1 \)

Step 1: Understanding the Inverse Function

To find the inverse function, we need to determine what input, \( x \), gives the output \( f(x) = 3 \). This requires us to search for \( x \) such that \( f(x) = 3 \) in the table.

Step 2: Find Value of \( x \) Corresponding to \( f(x) = 3 \)

Examine the table for \( f(x) \): \[\begin{array}{c|c|c|c|c|c} x & 0 & 1 & 2 & 3 & 4 \ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} \]From the table, locate the \( x \) value when \( f(x) = 3 \). In this case, \( f(1) = 3 \).

Step 3: Conclusion: Determine \( f^{-1}(3) \)

Since \( f(x) = 3 \) when \( x = 1 \), we conclude that \( f^{-1}(3) = 1 \). Inverse functions 'reverse' the input-output pairing, meaning we seek the original input when given an output.

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics, where we determine the output of a given function for specific input values. It's like asking a question and getting an answer. This is crucial for understanding how functions work and their behavior. Here's how to think about it:

• Every function, like \(f(x)\), takes an input \(x\) and provides an output \(f(x)\).
• In our problem, we're given a table with values for \(f(x)\).
• To evaluate \(f(x)\) means filling the role of the detective who finds the corresponding \(f(x)\) for each \(x\).

Imagine you are exploring the table to find the treasure that maps each input \(x\) to its mysterious outcome \(f(x)\). It's about effectively using the table to streamline this process. You don’t need to "solve" it through complicated math, but simply choose the value indicated. Evaluating a function, in essence, is reading and interpreting these tables correctly to understand which input produces the desired output.

Input-Output Relationship

The input-output relationship in functions explains how each input value \(x\) is allocated to an output \(f(x)\). This is like a factory: you put a raw material in (input), and a product comes out (output). Understanding this concept helps clarify how functions work:

• Each input \(x\) corresponds to exactly one output \(f(x)\), reflecting a one-to-one mapping in function tables.
• To determine what \(f^{-1}(3)\) is, we need to see which input in the table gives us the output 3.
• This process is essentially reversing this one-way street of function to find where a result came from.

By using the inverse function, we're walking back through this relationship. We start with an output and travel backward to identify the starting input value used to achieve it. In essence, this relationship is a cycle of finding connections and associations in a function's flow from input to output and back again.

Mathematical Tables

Mathematical tables serve as a visual aid that organizes data elements into rows and columns, helping simplify the process of function evaluation and inverse function determination. Here's how you can make the most of them:

• Each row typically represents the path of different inputs through the progression of outputs.
• By examining these tables, you can easily spot the relationships between inputs and their corresponding outputs.
• When dealing with inverse functions, like finding \(f^{-1}(3)\), tables allow you to quickly identify what input produced the output 3.

This graphical representation of functions can make complex operations seem simple. Instead of calculating, you access information instantaneously by interpreting the organized data points. Tables embody the bridge between learning theory and applying that theory practically, unraveling complexity through systematic arrangement. They turn function evaluation into a straightforward exercise of connecting dots neatly laid out on a page.