Problem 14 - Applied Calculus

Question

For the functions in Problems \(10-14,\) find \(f(5)\).$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\\hline f(x) & 2.3 & 2.8 & 3.2 & 3.7 & 4.1 & 5.0 & 5.6 & 6.2 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

\( f(5) = 4.1 \)

Step 1: Understand the Table

The given table associates each value of \( x \) with a corresponding value of \( f(x) \). Each \( f(x) \) is matched to an \( x \) value, such as \( f(1) = 2.3 \), \( f(2) = 2.8 \), and so on.

Step 2: Identify the Requested Input

The problem asks to find \( f(5) \). This means we need to locate the \( f(x) \) value that corresponds to \( x = 5 \) in the table.

Step 3: Extract the Value

From the table, when \( x = 5 \), the corresponding value of \( f(x) \) is 4.1. Therefore, \( f(5) = 4.1 \).

Key Concepts

Values Table

When working with functions, a values table is a crucial tool that helps organize pairs of inputs and outputs in a clear manner. Each row in the table directly connects a specific input value, also known as independent variable, with its corresponding output, or dependent variable. In the context of the given exercise, the values table lists the potential inputs, denoted as \( x \), with their matching outputs, \( f(x) \).

By examining this structured format, you can easily identify what the result will be for any given input. Such a table ensures that no values are left out and allows for quick retrieval of function results, such as looking up \( f(5) \) to find it equals 4.1. It's like a well-organized directory where each call has a defined response, ensuring a seamless function evaluation.

Input-Output Mapping

Input-output mapping is the backbone of understanding how functions operate. The idea is simple—a function takes an input, performs a certain set of operations or "rules," and produces an output. Just like a vending machine: you put in your money and select a snack (input), and the machine dispenses the snack (output).

In mathematical terms, we use notation like \( f(x) \) to denote the function, with \( x \) being the input, and \( f(x) \) being the resultant output. For instance, in the exercise provided, the input \( x = 5 \) maps to an output \( f(5) = 4.1 \). Every input has one, and only one, corresponding output, ensuring consistency and reliability in function behavior.

Mathematical Functions

The concept of a mathematical function is foundational in both basic arithmetic and advanced mathematics. A function can be thought of as a special kind of relationship where each input is linked to exactly one output. This relationship can be illustrated in several forms, including equations, tables, or graphs.

A function like \( f(x) \) covers any rule that assigns to each number \( x \) in a set, a unique number \( f(x) \). Functions are often expressed with formulas, but can also be represented in values tables, as seen in the exercise. Whether solving practical problems, modeling real-world situations, or performing complex calculations, understanding functions is key to interpreting how variables interact under different circumstances.