Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, use the values listed in to evaluate or solve. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 8 & 0 & 7 & 4 & 2 & 6 & 5 & 3 & 9 & 1 \\ \hline \end{array} $$ Find \(f(1)\).

Step-by-Step Solution

Answer

\(f(1) = 0\).

Step 1: Understanding the Problem

We need to find the value of the function \(f(x)\) when \(x = 1\) using the given table of values. The table lists values of \(x\) and their corresponding \(f(x)\) values.

Step 2: Locating the Value in the Table

Identify the row labelled \(x\) and find where \(x = 1\) appears. Then, move directly down to the row labelled \(f(x)\) to find the corresponding function value.

Step 3: Reading the Corresponding Value

When \(x = 1\), read the value in the \(f(x)\) row. According to the table, when \(x = 1\), \(f(1) = 0\).

Key Concepts

Function Table

Function tables are a helpful way to organize and display the relationship between a set of inputs and their corresponding outputs in a function. In a function table, each row typically represents a different input, and its related output, also known as the function value, is listed alongside it. The input values are known as the domain, while the output values make up the range of the function.
A function table is particularly useful when dealing with discrete functions, where inputs are separate and distinct values. This discrete arrangement enables easier identification and computation of function values.
To use a function table, follow these simple steps:

• Locate the input value needed, often found in the top row or column.
• Find the corresponding output value by following directly down or over to the aligned function value.

Function tables also make it easier to spot patterns or visualize how a function behaves across different inputs. This clarity is invaluable for students who are just getting familiar with the concept of functions.

Discrete Functions

Discrete functions are functions where the input values are distinct and separate from each other, as opposed to continuous functions that can have input values from a continuous range. With discrete functions, the inputs are usually whole numbers, which makes them easier to handle in terms of computation and graphing.
When dealing with discrete functions, such as those represented in the exercise table, you only evaluate the function at specific points. This specific evaluation is what makes the function discrete.
Here are some key points to understand about discrete functions:

• The inputs are usually finite and countable.
• The outputs can also be enumerated.
• A table is a handy tool to list all possible inputs and their corresponding outputs.

Discrete functions are the backbone of many real-world applications where measurements are taken at distinct intervals or where only whole numbers make sense, like counting objects or steps.

Function Mapping

Function mapping describes the process of pairing each element from a set of inputs, or domain, with exactly one element from a set of outputs, or range. This concept helps students understand how functions operate and is crucial in a variety of mathematical analyses.
In essence, a function acts like a rule or machine. You "input" a value, and it "maps" this value to an "output" according to this rule or relation. In the case of our table, the function mapping is visually illustrated by the corresponding rows for each input-output pair.
Points to remember about function mapping include:

• Each input has exactly one output, ensuring a clear and non-ambiguous mapping.
• Function mapping helps visualize the relationship between variables.
• It's a foundational principle in algebra and calculus.

By fully understanding how function mapping works, students gain a deeper insight into how various functions can be represented and analyzed, which is key to mastering advanced math concepts.