Problem 66 - College Algebra with Corequisite Support

Question

For the following exercises, use the function \(f\) represented in the table below. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 74 & 28 & 1 & 53 & 56 & 3 & 36 & 45 & 14 & 47 \\ \hline \end{array} $$ Evaluate \(f(3)\).

Step-by-Step Solution

Answer

The value of \(f(3)\) is 53.

Step 1: Locate x in the table

Find the value of \(x\) that you need to evaluate in the function table provided. We are asked to find \(f(3)\), so look for the column where \(x = 3\).

Step 2: Find corresponding f(x) value

After locating \(x = 3\) in the table, look directly below it to find the corresponding value of \(f(x)\). According to the table, when \(x = 3\), \(f(x) = 53\).

Step 3: Confirm result

Double-check the values from the table to ensure the correct value for \(f(3)\). We confirm that \(f(3) = 53\) from the table.

Key Concepts

Function Table

A function table is an easy way to organize and display the values of a function. It consists of two main rows: one for the input values, commonly labeled as \(x\), and another for the output values, labeled as \(f(x)\). This setup allows you to quickly find the output for a given input.
Think of the function table as a translator. It tells you how each input \(x\) corresponds to an output \(f(x)\).

• The top row displays your inputs, typically starting from \(x = 0\) and increasing sequentially.
• The bottom row directly underneath provides the output for each input.

Using this table can simplify the process of evaluating functions, as all of the information you need is in one place.

Input-Output Relationship

In any function, understanding the input-output relationship is essential. The relationship describes how the input values, \(x\), are transformed into output values, \(f(x)\).
Functions act like a machine or a box where you "input" a value and after processing, the "output" comes out. Each different input results in a specific output based on the rule defined by the function.

• The input-output relationship is predictable and consistent for a given function.
• A function table showcases the specific outputs that result from each input.

This relationship is fundamental to functions, as it helps us to understand their behavior and predict the outcomes.

Evaluating Functions

Evaluating functions is the process of finding the output (or result) for a specific input using the function rule or table. This process may seem tricky at first but becomes straightforward once you get the hang of it.
To evaluate a function:

• Identify the input value you are working with; in our example, it is \(x = 3\).
• Refer to the function table to find which output corresponds to your input; here, \(f(3) = 53\).
• Always confirm your findings by double-checking the table or calculations.

Through these steps, you can determine the output for any given input, making function evaluation a predictable and reliable process.